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THE GLASGOW TEXT BOOKS OF CIVIL 

ENGINEERING. Edited by G. MONCUR, B.Sc. 

M.I.C.E., M.Am .Soc.C.E. Professor of Civil Engineer- 

ing in the Royal Technical College, Glasgow. 



HYDRAULICS OF PIPE LINES 



THE GLASGOW TEXT BOOKS. 
Edited by G. MONCUR. 



HYDRAULICS OF. 
PIPE LINES 



k BY 

W: F. DURAND, Ph.D. 

PROFESSOR OF MECHANICAL ENGINEERING. STANFORD UNIVERSITY, CALIFORNIA 



With 134 Illustrations 




NEW YORK 

D. VAN NOSTRAND COMPANY 

EIGHT WARREN STREET 

1921 



G 



1 * 



Printed in Great Britain 



] \ L I 



11' ' y7 l 



AUTHOR'S PREFACE 

This book is intended to give, in a reasonably adequate engineering 
form, a discussion of the more important hydraulic problems which 
arise in connection with pipe lines and pipe line flow. No attempt 
has been made to cover the subject from the structural or descrip- 
tive viewpoints. In Chapter V the treatment is partly structural 
and descriptive, but rather as incidental to the main purpose of the 
work. 

Chapter I presents briefly the elementary principles relating to 
pipe line flow with some special emphasis on the subjects of pipe 
friction and secondary losses due to miscellaneous turbulence. 
Special attention may also, perhaps, be directed to the series of 
relations and expressions in Section 11 and to the treatment of 
network systems in Section 21. 

Chapter II presents the subject of surge, not with analytical 
detail but in such manner as to place before the engineer a variety 
of means for dealing with this important problem. Attention may 
be directed to the special application of the principles of similitude 
as applied to this problem, thus bringing it within the range of 
laboratory investigation. 

In Chapter III is given a reasonably full analytical treatment of 
the subject of water ram or shock. This seems to be justified by 
the absence, so far as the author is aware, of any measurably 
adequate discussion of this subject in the English language. The 
method employed starts with the fundamental principles as first 
developed by Joukovsky. The details of the development are 
however, largely independent of other sources. Special attention 
may be called to the discussion of other proposed formulae for 
shock, showing the necessary limitations which must surround 
their use, and in particular to that of Allievi which has been so 
commonly employed without a proper appreciation of its necessary 
limitations. Attention may also be called to the extension of this 
method to include the hypothesis of partial or imperfect reflection 
at the valve — and of the need of experimental work to serve as a 



viii AUTHOR'S PREFACE 

basis for determining what degree of reflection may be expected 
under various operating conditions. 

Chapter IV presents, with some new material, the general subject 
of stresses in pipe lines, due either to static pressure or under 
conditions of flow. 

Chapter V presents some descriptive and structural material, 
especially with reference to materials, joints, fastenings, fittings, 
etc. It gives likewise some discussion of the problem of economic 
design and of a number of special problems connected with the 
installation and equipment of pipe lines. 

Chapter VI presents a brief discussion of oil pipe lines, or more 
broadly of any pipe line intended especially for the carriage of 
viscous fluids. Particular attention may be directed to the method 
of treatment involving the use of the general equation of flow as 
discussed briefly in Appendix I. There seems good reason to 
believe that if the frictional resistance is thus determined, as a 
function of diameter, velocity, density and viscosity, with a proper 
allowance for the physical condition of the flow surface of the pipe, 
the results will be entirely reliable and the design of such pipe lines 
may be undertaken with the same degree of confidence as in the 
case of those for the flow of water. 

Any work of the character of this volume must be, in large parts, 
a compilation and adaptation of material drawn from many sources. 
It has been the intention to give credit whenever direct use has thus 
been made. The reader will find, however, a considerable amount 
of material which is either new or which has not been commonly 
presented in the form here given. 

It is hoped that the work may prove of some direct help to 
engineers in dealing with the hydraulic problems of pipe fine flow, 
and also that it may serve as a stimulus to the further study of 
many phases of these problems regarding which our knowledge is 
still entirely too fragmentary. 



CONTENTS 



CHAPTER I 

GENERAL HYDRAULIC PRINCIPLES — STEADY MOTION 

PAGE 

Energy of flowing stream — Loss of head — Distribution of velocity and 
mean velocity head over cross section of pipe — Hydraulic gradient 
— Capacity — General problem of steady flow — Free surface flow — 
Line connecting two reservoirs — Power delivered at discharge 
end — Piping systems . . . . .... . . . . . . 1 

CHAPTER II 

THE PROBLEM OF THE SURGE CHAMBER 

General statement of problem — Derivation of equations — Modes of 
treatment — Special cases — Approximate general solution — Prob- 
lem as simplified by disregard of governor action — Treatment by 
methods of approximate integration — Treatment through assump- 
tion of predetermined acceleration history for water — Treatment 
by law of similitude — Differential surge chamber . . . . . . 59 

CHAPTER III 

WATER RAM OR SHOCK IN WATER CONDUITS 

Shock with assumed instantaneous complete closure — Physical condi- 
tions — Modifications necessary to allow for elasticity of pipe and 
compressibility of water — Velocity of propagation of acoustic 
wave — Excess pressure developed — Analytical treatment for non- 
instantaneous movement of valve, opening or closure, complete 
or incomplete — Treatment with assumed partial reflection at valve 
while partly open — Discussion of formulae with numerical cases — 
Approximate formulae, discussion and limitations of use . . . . 84 

CHAPTER IV 

STRESSES IN PIPE LINES 

Ring tension — Longitudinal stress — Stresses due to angles, bends and 
fittings under static conditions — Same under flow conditions — 
Load due to weight of pipe or element, also of contained water — 



CONTENTS 



PAGE 



Stresses in expansion joints — Influence of anchors, piers, ties, 
abutments, etc. — Flat plates supported by ribs — Joint fastenings 
— Bending moment in spans — Combined stresses . . . . . . 159 

CHAPTER V 

MATERIALS, CONSTRUCTION, DESIGN 

Pipe materials — Commercial pipe, steel and cast iron — Joints — Con- 
nections — Wood stave pipe — Reinforced concrete pipe — Design 
— Economic dimensions — Expansion and contraction due to 
changes of temperature, erection of pipe lines — Piers and anchors 
— Protective coatings — Air relief valves — Manholes and covers 
— Expansion joints — Pipe line fittings . . . . . . 190 

CHAPTER VI 

OIL PIPE LINES 

Physical properties of oil affecting pipe line flow — Oil pipe lines — 

Design of general characteristics . . . . . . . . . . 246 



APPENDICES 

I. General theory of pipe line flow m 257 

II. Expression for F (Chapter III) in terms of v and e . . • . 263 

III. Force reaction between element of a conduit and a flowing stream 264 

IV. Economic design . . . . . . . . . . . . • • 266 



pipe 



LIST OF ILLUSTRATIONS 

FIG. PAGE 

19 

19 

19 
20 
22 
22 
22 
23 
32 
34 
35 
35 
36 
37 
39 
41 
44 
47 
48 
51 
52 
55 
57 
60 
62 
62 



1. Forms of inlet 

2. Loss due to enlargement 

3. Loss due to contraction 

4. Loss due to obstruction 

5. Loss due to gate valve . 

6. Loss due to butterfly valve 

7. Loss due to plug cock . 

8. Loss at elbow 

9. Variable flow over cross section of 

10. Annular elements of cross section 

11. Integration for volume flow 

12. Integration for energy flow 

13. Hydraulic grade . 

14. Hydraulic grade 

15. Flow in circular section partially filled 

16. General problem of pipe line flow .... 

17. Flow in open channel ...... 

18. Flow in conduit between connecting reservoirs . 

19. Flow in line partly with free surface and partly with full section 

20. Branching system ....... 

21. Single Y branch ....... 

22. Branching system . . 

23. Branching system with double supply 

24. Problem of the surge chamber ..... 

25. Acceleration head ...... 

26. Retardation head ....... 

27. Time history of movement of water level, velocity and acceleration 

head ........ 

28. Complete shut-down with overflow 

29. Complete shut-down without overflow 

30. Diagram of movement of water level and velocity (opening) 

31. Diagram of movement of water leveFand Velocity (closure) 

32. Rule for approximate integration .... 

33. Similar curves ....... 

34. Shock in pipe lines ...... 

35. Successive states in oscillating water column 

36. Successive states in oscillating water column 

37. Time history of excess pressure at valve. Ideal case 

xi 



67 
70 

70 
71 

72 
78 
80 
85 
85 
86 
87 



xii LIST OF ILLUSTRATIONS 

FIG. PAGE 

38. Time history of excess pressure at points in the line. Ideal case 87 

39. Time history of excess pressure at points in line. Ideal case . 88 

40. Block model for time history of excess pressure at any point in line. 

Ideal case . . . . . . 88 

41. Excess pressure as modified by effects due to friction. Partial 

account ........ 90 

42. Excess pressure as modified by effects due to friction. Partial 

account . . . . . . . 92 

43. Excess pressure as modified by effects due to friction. Partial 

account ........ 92 

44. Excess pressure as modified by effects due to friction. Partial 

account . ........ 93 

45. Time history of dv/dt and v . . . . .99 

46. Graphical solution of equation giving value of pressure head h . 118 

47. History of pressure head h assuming uniform rate of velocity 

change (closure) . . . . . . .120 

48. History of pressure head h and velocity v (closure) . 129 

49. History of pressure head h and velocity v (closure) . . . 129 

50. History of pressure head h (closure) . . . . 130 

51. History of pressure head h and velocity v (closure) . . .131 

52. History of pressure head h and velocity v (closure) . 132 

53. History of pressure head h showing course after full closure . 133 

54. History of pressure head h after arrest of valve movement at -6z 

(closure) . . . . . . . . . 133 

55. History of pressure head h after arrest of valve movement (closure) 134 

56. History of pressure head h after arrest of valve movement (closure) 134 

57. History of pressure head h (opening) . . . . . 135 

58. History of pressure head h after arrest of valve (opening) . .135 

59. History of pressure head h after arrest of valve (opening) . . 136 

60. History of pressure head h after arrest of valve (opening) . .137 

61. History of pressure head h after arrest of valve (opening) * . 137 

62. History of pressure head h after arrest of valve (opening) . .138 

63. History of pressure head h after arrest of valve (opening) . . 138 

64. History of pressure head h with increase in value of T (opening) . 139 

65. History of pressure head h (opening) T>z . . . 139 

66. History of pressure head h (opening). Influence of increasing T 139 

67. History of pressure head h (opening). Influence of increasing T 140 

68. History of pressure head h (opening). Influence of increasing T 140 

69. History of pressure head h (opening) . . . . .141 

70. History of pressure head h (opening) . . . . 141 

71. History of pressure head h starting from partial opening . .141 

72. History of pressure head h, starting from partial opening . . 142 

73. History of pressure head h, starting from half opening . . 142 

74. History of pressure head h, starting from half opening . . 143 

75. History of pressure head h, starting from half opening . .143 

76. History of pressure head h for various points in pipe line (closure) 144 

77. History of pressure head h for various points in pipe line (closure) 144 

78. History of pressure head h for various points in pipe line (closure) 145 



LIST OF ILLUSTRATIONS 



Xlll 



PIG. 

79. History of pressure head h for various points in pipe line (opening) 

80. History of pressure head h for various points in pipe line (opening) 

8 1 . History of pressure head h with assumed partial reflection at valve 

(closure) ........ 

82. History of pressure head h with assumed partial reflection at valve 

(closure) . . . . . . . 

83. History of pressure head h with assumed partial reflection at valve 

(closure) . . 

84. History of pressure head h with assumed partial reflection at valve 

(closure) . . . . . . . 

85. History of pressure head h with assumed partial reflection at valve 

(opening) ........ 

86. History of pressure head h with assumed partial reflection at valve 

(opening) . . . . . ... 

87. Resultant force on pipe line elements — static conditions 

88. Resultant force on pipe line elements — static conditions 

89. Distribution of resultant load along pipe line bend 

90. Resultant force on pipe line elements — static conditions 

91. Resultant force on pipe line elements — general case with flow . 

92. Resultant force on pipe line elements — flow from open reservoir 

93. Resultant force on pipe line elements — flow from open reservoir 

94. Resultant force on pipe line elements — case with multi -discharge 

95. Resultant force on pipe line elements — discharge from open reser 

voir through elbow and nozzle . 

96. Expansion joint 

97. Expansion joint 

98. Expansion joints combined with elbow 

99. Resultant force on long elastic pipe . 

100. Resultant force on long elastic pipe 

101. Resultant force on long elastic pipe . 

102. Reaction on support 

103. Reaction on support 

104. Reaction on support 

105. Reaction of elbow on flexible ties 

106. Reaction of elbow on direct support 

107. Reaction of elbow on constraining piers 

108. Stress on elliptical section 

109. Bell and spigot joint . 

110. Special forms ball and socket joints 

111. Universal pipe joint 

112. Riveted joints 

113. Riveted joints for longitudinal seams 

114. Riveted joint for longitudinal seams 

115. Riveted joint for circumferential seams 

116. Riveted joint for circumferential seams with spun lead filling 

between ends of sections 

117. Forms of flange joints . 

118. Forms of flange joints 



PAGE 

145 

146 

146 

147 

147 

148 

148 

149 
162 
163 
163 
165 
166 
167 
168 
169 

170 
173 
173 
175 
176 
176 
178 
181 
182 
183 
183 
183 
183 
186 
192 
193 
193 
196 
197 
197 
200 

201 

204 
204 



XIV 



LIST OF ILLUSTRATIONS 



PIG. 

119. Forms of flange joints . 

120. Form of flange joint 

121. Section of wood stave pipe 

122. Saddle for round rod ties — wood stave pipe 

123. Design of thickness for shell of pipe line 

124. Layout of thickness on pressure head 

125. Layout of thickness on pressure head 

126. Diagram for determination of thickness of pipe line shell 

127. Saddles for support of pipe line 

128. Design of air relief valves . 

129. Design of air relief valves 

130. Automatic pressure relief valve . 

131. Testing bulkhead 

132. Oil pipe line design — graphical construction 

133. Diagram of values of / on abscissa of Dw//t. 
134.. Economic value, graphical determination 



PAGE 

205 
205 
206 
206 
213 
221 
222 
223 
229 
235 
239 
241 
244 
255 
258 
267 



PRINCIPAL NOTATION EMPLOYED 



A — Area of pipe or conduit. 
a — Area of nozzle or through discharge valve. 

B — Summation of terms following first in equation (42) Chapter III. 
b — Head due to atmosphere. 
b — Expression L/C 2 r. 
O — Chezy coefficient, 
c— Expression L/C 2 r+l/2g. 
D — Diameter of pipe or conduit. 
d — Diameter of pipe or conduit, 
e — Efficiency of riveted joint. 
F — Area of surge chamber. 
/ — Efficiency of nozzle. 
/ — Friction coefficient in Darcy formula. 
H — Total head above atmosphere or special pressure datum. 
h — Lost head due to friction and turbulence. 
h — Change of pressure head (increase or decrease) attendant on closure 

or opening of valve at discharge end of line. 
i — Hydraulic gradient. 
t — Time x/S, section (38). 
K — Coefficient of elasticity of water (cubical), Chapter III. 
k — Ratio m/T in case of valve closure or opening, as in section (42). 
L — Length of pipe line or conduit. 
M — Bending moment. 
M — Momentum. 
m — Ratio of area of nozzle or through valve to area of pipe or ratio between 

areas of different parts of a complex pipe line. 
n — Kutter's roughness coefficient. 
p — Pressure intensity in general. 
q — Pressure change (increase or decrease) attendant on closure or opening 

of valve at discharge end of line. 
r — Mean hydraulic radius. 
S — Velocity of propagation of acoustic wave. 
8 — Change in velocity v attendant on opening or closure of a valve in 

delivery end of line. 
T — Tensional stress. 

T — Time of valve movement (opening or closing) at delivery end of pipe 
line. 
t — Time in general. 
t — Thickness of pipe wall. 



xvi NOTATION FOE UNITS OF MEASURE 

u — Velocity at mouth of nozzle. 
V — Volume in general, usually in f 3. 

v — Velocity of flow in pipe line or conduit, usually in fs. 
w — Density of water. 
x — Variable length in general. 
y — Head due to some special pressure. 
Z — Total head, including atmosphere or special pressure. 
z — Elevation above datum level. 

z — Time 2L/S required for an acoustic wave to travel twice the length of 
the line L. 

a — Ratio S/g or velocity of acoustic wave divided by gravity. 

/i — Coefficient of viscosity. 

a — Density of steel, or (Chapter VI) density in general. 

Other notation employed as convenient and as defined in connection 
with special problems. 



NOTATION FOR UNITS OF MEASURE 

In order to facilitate representation in typographical form the following 
notation is employed : 

Feet (f ) 

Inches (i) 

Square feet (f2) 

Square inches (i2) 

Pounds per square foot (P*2) 

Pounds per square inch (pi2) 

Pounds per cubic foot (pf3) 

Seconds (s) 

Feet per second (fs) 

Cubic feet per second (f3s) 

Note that these designations in Roman type are placed in parentheses. 
This will serve to distinguish them from other quantities which might be 
denoted by the same letters. 

Note also that throughout the text the oblique line is used as the sign of 
division, thus a/b, dv/dt, etc. 



HYDRAULICS OF PIPE LINES 

CHAPTER I 

GENERAL HYDRAULIC PRINCIPLES— STEADY MOTION 

i. Energy of a Flowing Stream 

As shown in the elementary theory of hydraulics the total energy 
of a flowing stream must be considered under three heads. Specific- 
ally in referring to the energy of a stream we mean the energy of 
one pound of the liquid at a definite location in the fine. In 
hydraulics the word head is very commonly used instead of energy. 
It must be remembered, however, that the word head thus em- 
ployed means in effect the energy of one pound of the liquid 
contents of a flowing stream. With this understanding the three 
components of the energy or head are as follows : 

(a) Pressure energy or pressure head — 

v 2 

(b) Kinetic energy or velocity head =- 

(c) Potential energy or gravity head z. 

Where p= pressure, absolute (pf2). 
w= weight (pf3). 
v== velocity (fs). 
z= elevation of given point above reference datum (f). 

In addition to these three primary forms of energy we must 
recognize two incidental or secondary forms. 

(d) The kinetic energy of eddies, vortex motion and turbulence 

generally. 

(e) Heat energy. 

The entire theory of hydraulics, with special reference to the 
dynamics of stream and pipe fine flow, is based on certain funda- 
mental propositions which will be briefly stated, referring the 
reader to elementary textbooks on the subject for a more detailed 
treatment. 

1. The three fundamental forms of energy (a), (b) and (c) are 
mutually convertible into each other and into mechanical work. 

H.P.L.— B 



2 HYDKAULICS OF PIPE LINES 

2. Any and all of the three fundamental forms (a), (b) and (c) 
are convertible into either of the secondary forms (d) (e). 

3. Neither of the forms (d) or (e) is convertible into (a), (b) or 
(c), or into mechanical work. A vortex or eddy or turbulence of 
any kind once formed can never be untangled or transformed 
back into any of the three available forms. Heat once formed 
as a result of the deformation of stream line flow cannot be trans- 
formed directly back into any of the available forms (a), (b) or (c). 

4. Form (d) inevitably degenerates into (e) and becomes 
dissipated as such. 

5. Any action in a stream which results in the formation of 
energy in either of the forms (d) or (e) involves inevitably the 
ultimate dissipation of such energy as heat, and therefore the 
irreversible transfer of energy from an available form (a), (b) or 
(c) to a form absolutely unavailable in a mechanical sense. 

One of the most important of hydraulic problems is concerned 
with the loss of total available energy or head occasioned by or 
incidental to the transfer of a liquid through a pipe line or other 
form of conduit. 

Let Z denote the total available head or energy of one pound 
of water at the origin or starting point 0. At any other point P 
in the line let h denote the amount of energy which has been 
transferred from forms (a), (b) and (c) into forms (d) (e). 
Let Z denote the total available energy at P. 
Then from the conservation of energy we must have 
Z =Z+h 
or Z=Z Q — h. 

Again, if we represent Z by the sum of the three components 
(a), (b), (c) we have 

z °=u,+i+ z+h « 

All quantities on the right refer to the point P. The first three 
comprise the total available energy at P while the fourth represents 
the loss in available energy between the origin and P. The 
quantity h thus transferred irreversibly from the available energy 
to the unavailable form is called the loss of head. 

In this fundamental equation, as noted, p denotes the total or 
absolute pressure at any given point in the stream. In hydraulic 
formulae generally, p is more commonly used to denote rather the 
pressure above the atmosphere, and similarly pjw to denote the 
pressure head in excess of the atmospheric head. Care should be 
exercised in all cases to note the exact sense in which the symbol 
for pressure is employed. 

If h is neglected in equation (1) we have the well-known Ber- 
nouilli equation for steady flow without loss of head. Including the 
term h, (1) is to be considered as the general energy equation for 



GENERAL HYDRAULIC PRINCIPLES 3 

one pound of water at any point in the line and in which the first 
three terms denote the total available energy of one pound of water 
at the point P, while h represents the work done against various 
forms of resistance (and hence the energy transferred from the 
available forms (a), (b), (c) into the unavailable forms (d), (e)) in 
carrying one pound of water from to P. 

More briefly h may be viewed as the measure of the work done 
against secondary and viscous forces in carrying one pound of 
water from to P. 

Loss of head or loss of energy may thus be viewed directly as 
the energy equivalent of the work involved in various accidental 
phenomena connected with stream line flow. Thus wherever the 
lines of stream flow are abruptly diverted or redistributed, or 
whenever the stream undergoes abrupt changes in size, or whenever 
stream flow undergoes abrupt change in geometrical character 
generally, there is always, in a slightly viscous liquid such as water, 
a tendency toward the formation of eddies, vortices and confused 
turbulence. Such confused turbulence is also formed at the surface 
of separation between a flowing stream and the containing conduit, 
or generally at the surface of separation of a solid and a liquid in 
relative motion. The confused turbulence thus formed at the 
surface of the containing conduit is due partly to the action of 
roughness and irregularity of surface and partly to the action of 
adhesive forces acting between the solid and the liquid. 

In all cases, and no matter how produced, the formation of 
eddies, vortices and turbulence in general requires the expenditure 
of work which can only be furnished by transfer from the available 
forms (a), (b), (c). Such formation always involves, therefore, a 
transformation of available energy into forms (d) and (e) ; primarily 
into (d) and ultimately all into (e). 

For convenience of discussion we may classify these losses as 
follows : 

(1) Loss of head due to turbulence formed at or near the surface 

of the containing conduit and due to roughness of surface 
and to the action of adhesive forces. 

(2) Loss of head due to all other causes involving miscellaneous 

turbulence and redistribution of stream line flow. 

Loss (1) is commonly referred to as due to friction, though the 
actual phenomena involved have little in common with those 
characterizing the frictional resistance between two solids. 

Loss (2) we shall refer to as due to miscellaneous turbulence. 

Before proceeding with the discussion of these various forms of 
loss of head it will be well to note at this point that in Appendix I 
will be found a brief discussion of the general theory of pipe line 
flow as developed from the principle of dimensions, and including 
the influence due to the viscosity and density of the liquid, and with 
due regard to their dependence in turn on temperature. It appears 



4 HYDRAULICS OF PIPE LINES 

that there are in general two modes of flow for a fluid moving in a 
pipe or conduit : (1) stream line and (2) turbulent, but that in all 
cases arising in ordinary engineering practice, the flow of water 
occurs under the turbulent mode. 

With regard to viscosity, the variation in the case of water, at 
least over ordinary working temperatures, is relatively small, and 
viscosity as a factor in the problem is of relatively small importance 
in the turbulent mode of flow. For these reasons, in ordinary 
hydraulic problems, commonly no attempt is made to include this 
factor. 

With regard to density, the variation with temperature is 
relatively greater, but the influence of this factor on lost energy 
measured in terms of head is relatively small in the case of turbulent 
flow, and it is therefore commonly neglected. If the loss is expressed 
in terms of pressure, however, the density enters directly as a factor 
and care must be taken in transforming head into pressure that the 
proper value of the density as effected by the temperature is 
employed. 

On the other hand, in the case of crude petroleum oil and such- 
like liquids, the influence of viscosity as dependent on density and 
temperature commonly plays a controlling part and must there- 
fore be included in any determination of the value of the lost head. 

For a discussion of formulae and methods of computation where 
the influence of viscosity must be included, see Appendix I and 
Chapter VI on oil pipe lines. In the present chapter, dealing with 
water conduits primarily, the usual practice will be followed and the 
relatively small influence due to variations of viscosity and density 
with temperature will be neglected. 

We shall now proceed with the discussion of certain formulae 
commonly used for the computation or estimation of these various 
forms of loss of head. 

In general we shall designate all such losses by h. The context 
will always show whether special reference is intended to loss (1) as 
above (skin effect) or loss (2) (turbulence) or to the sum of the two. 



2. Loss of Head. Chezy Formula 

The loss of head due to so-called " fluid friction " in a conduit 
with liquid flowing under steady conditions depends, aside from 
viscosity and density, on the following factors : 

(a) The velocity of flow. 

(b) The character of the surface. 

(c) The length of the conduit. 

(d) The transverse dimensions and form of the conduit. 

Various formulae have been proposed for the purpose of relating 
the factors in this problem. The best known and most commonly 
employed is the so-called Chezy formula. 



GENERAL HYDRAULIC PRINCIPLES 5 

Let v=vel. (is). 
C— coefficient. 

i— hydraulic gradient =h/L. 

r=hydraulic mean radius =A /P. 

ft = friction head (f). 
L=length of conduit (f). 
A= cross section area (f2). 
P— wetted perimeter (f). 

We then have the following formulae : 

v=Cs/Tr (2) 

or h== C*r () 

In this formula for h, which is admittedly not fully rational, 
especially as regards the index of v, the coefficient C must include 
some influence due to factors (a), (b) and (d) above. With values 
drawn from experience, however, and representing these factors, 
the formula will give reliable results within the range of values 
covered by the experimental basis. 

It results that the values of C in this formula must be selected 
with reference to velocity, roughness, and size of conduit. 

In the use of the Chezy formula or any of its derivatives, care 
must be taken to distinguish between the hydraulic gradient i and 
the actual gradient on which the pipe is laid. In the case of a pipe 
running under pressure, there is no necessary relation between 
the two. 

3. Loss of Head. Kutter's Formula 

As an aid in the selection of a value of G for the Chezy formula, 
Kutter's formula is frequently employed, though it should not be 
forgotten that this formula was orginally developed for the dis- 
cussion of the problem of the flow in open channels. The formula 
is furthermore empirical rather than rational in its relation of the 
three controlling variables, hydraulic gradient, roughness and 
geometry of conduit. The use of this formula for the determination 
of the coefficient C in the Chezy formula (equation (3)), should 
therefore be made with some reserve, and preferably as based on 
observation for closely similar conditions. 

Let Thoroughness coefficient. 

Then in English units Kutter's formula is as follows : 

LSI [23+1^+11 
0= t I M (4) 

In this formula the hydraulic gradient i is intended to represent 
the velocity, r represents the transverse dimensions or cross section 



6 HYDRAULICS OF PIPE LINES 

of the conduit and n represents the character and condition of the 
surface. 

In any given case i and r will be known and n must be assumed 
by judgment in accordance with the character of the surface. 
Typical values will be found in Sec. 7. 

The computation of values from (4) is somewhat tedious, and 
examination shows further that the variation over the range of 
ordinary values of i is relatively small, and in consequence by taking 
i constant at a mean value, the formula may be simplified in marked 
degree. Thus if i be taken constant at 1 : 1000 the formula reduces 
to i.g 

—+45 

C=^— (5) 

The simpler form of (5) may be further justified by the considera- 
tion that the presumable error due to the use of (5) rather than (4) 
will be small in comparison with the uncertainty which will inevit- 
ably attach to the arbitrary selection of a value for the roughness 
coefficient n, or to the use of the formula itself for finding the value 
of C. 

4. Loss of Head. Exponential Formula 

Reference has been made in Sec. 2 to the fact that the Chezy 
formula is not quite rational in taking resistance or loss of head to 
vary with the square of the velocity. 

Experiments with varying velocities in conduit flow as well 
as a vast amount of research in the related field of ship resistance 
show that, with the other factors constant, the resistance or loss 
of head due to friction varies with the speed according to an index 
which may range about 1-83, but which, in practically all cases, is 
less than 2. In a more fully rational formula for loss of head, the 
velocity v should, therefore, have an index approximately 1-83 or 
1-85. Several formulae of this character have been proposed. Of 
these one of the best known is that proposed by Williams and 
Hazen, as follows : 

Let r=mean hydraulic radius=-4/P as in Sec. 2. 
i=hydraullc gradient =h/L as in Sec. 2. 
J5= coefficient. 
0= coefficient in Chezy formula. 

Williams and Hazen Formulce : 

v== Br°' Q H°- 54 > (6) 

Lv 185 
and ft- ffi.85,.1.17 P) 

or v=l-318Cr 88 r 54 (8) 

Lv 1 ' 85 

and A =^66 8(7i»V" (9) 



GENERAL HYDRAULIC PRINCIPLES 7 

In (6) and (7) B is a coefficient which is supposed to depend on 
roughness alone and which may be selected on the basis of the 
same features as for n in Kutter's formula. In (8) and (9) the 
numerical factor is introduced in order to relate B of (6) and 

(7) to the coefficient C of the Chezy formula. For average or 
typical values of i=-001, and r=l, it is readily seen that B= 
(•001)- 04 C=l-3180. Hence in the form of (8) or (9) C may be 
selected for the given character of surface and assuming a hydraulic 
gradient of -001 and a mean hydraulic radius of 1. This value in 

(8) or (9) will then give the proper results with the actual gradient i 
and actual radius r. The significance of this form of the coefficient in 

(8) and (9) lies in the fact that many engineers prefer to estimate 
directly the value of G in the Chezy formula, and, with nearly 
standard conditions as to hydraulic gradient and radius, are able 
to do so with quite satisfactory accuracy. There has accumulated 
in the literature of this subject, furthermore, a large amount of 
material bearing directly on values of G for the Chezy formula, and 
thus serving as a basis for the immediate selection of this coefficient 
for various conditions. If it is understood that the C of (8) and 

(9) is the G of the Chezy formula for r=l and t = -001, a large 
amount of this data becomes immediately available for purposes of 
selection. 

The principal practical drawback to the use of exponential 
formulae lies in the more complicated numerical procedure which 
is involved. As an aid in making such computations effective use 
may be made of logarithmic cross section paper, and as an exten- 
sion of the same idea a special slide rule has been devised for the 
direct computation of the terms in equation (8). 

5. Loss of Head. Darcy's Coefficient 

The following formula is given in all elementary works on 
hydraulics : r w , 

h =4i < io > 

where /= coefficient as below : 

L=length. 
D= diameter. 
v= velocity (fs). 
Note that in the above formula L and D must both be measured 
in terms of the same unit. 

In this formula / represents a roughness or friction coefficient 
which decreases with increase in either D or v. The variation with 
v is, however, slight and may usually be neglected. For variation 
with diameter Darcy recommends a formula which may be 
expressed in the form nm fifi 

f=-02+'^p (11) 

where D is diameter in feet. 



8 HYDKAULICS OF PIPE LINES 

This value of / is recommended for clean pipe. For old pipe a 
suitable increase up to double value should be made. 

Comparison with the Chezy formula in Sec. 2 develops immedi- 
ately the following relations between / and C. 

8flr_257-3 




_ 16-05 
From the above value of / we readily find 



v, 



,12) 



•02D+-00166 

Corresponding values of / and C are given in Table I. 

6. Loss of Head. Volume Flow Formula 

Let L=length of conduit (f ). 
v= velocity of flow (fs). 
V— volume rate of flow (f3s). 
B= constant depending on form of cross section. 
0= coefficient in Chezy formula as in Sec. 2. 
r=hydraulic mean radius (f). 
-4=area of section of conduit (f2). 
Then we may put 

A=Br* 
and the Ch6zy formula 

, Lv* 

is readily put in the form 

LV 2 

B 2 C 2 r 5 (13) 

For a circular cross section B—4:7r, r—Dlk, and we have 

tt 2 C 2 D 5 ~ C 2 D 5 ( ' 



or 



v /.154MPh (15) 



7. Suggestions Regarding Practical Values 
of n and G 

The value of the roughness coefficient n of Kutter's formula, in 
cases arising in practice, is commonly found between -010 and -015. 



GENERAL HYDRAULIC PRINCIPLES 



9 



Following are values relating to pipe line or conduit surfaces as 
assigned by Kutter : 

Material. n. 

Well-planed timber -009 



Neat cement . 

Cement with one -third sand 

Unplaned timber . 

Ashlar and brickwork 

Unclean surfaces in sewers and conduits 
Later observations seem to indicate further values as follows : 
Concrete conduits with smooth plastered 

surface when new .... 

The same conduits with ageing and a gradu- 
ally acquired gelatinous or slime -covered 

surface ...... 

Iron and steel pipes ..... 

Wood stave pipe ..... 



•010 
•Oil 
•012 
•013 
•015 



•011 to -012 



•013 to -014 
•013 to -015 
•011 to -012 



There is, of course, no definite upper limit for the measure of 
roughness and with flaking and scaling of cement -lined surfaces or 
with corrosion and tuberculation of iron and steel pipe the values 
of n may rise to -020 or more. 

Turning now to the direct estimate of the value of C in the Chezy 
formula, the following suggestions are given : 

For new cast-iron pipe with specially smooth surface, C may 
rise to values approaching 140. For average new cast-iron pipe a 
value of about 130 may be anticipated. With corrosion and fouling 
the value of C will fall, according to the degree of roughness, to 
values approximating 100 or less. Where the capacity of a cast-iron 
pipe line after some period of years is in question, values of C from 
100 to 110 may usually be employed. 

For cast-iron pipe 4 inches diameter to 60 inches diameter, 
Williams and Hazen* estimate that beginning with 130 when new 
the value of G will decrease to 100 in from 13 to 20 years and to 80 
in from 26 to 47 years. In each case the shorter range of years is for 
the 4 -inch size and the larger range for the 60 -inch size. Inter- 
mediate ranges are for intermediate sizes, the range for any one 
value of C increasing at first rapidly and then more slowly in going 
from the smaller to the larger sizes. Again for each size of pipe the 
decrease in C is at first rapid and then more gradual with increasing 
age, the values for small pipes falling off more rapidly than those for 
large. These relations between the value of C, size of pipe and age 
are expressed in tabular form by the authors above mentioned, not 
as exact or precise results to be anticipated in all cases, but as a 
generalization based on wide observation and careful judgment. 
The authors are careful to state that the ranges of years stated are 



* "Hydraulic Tables," John Wiley and Sons, New York. 
Hall, London. 



Chapman and 



10 HYDKAULICS OF PIPE LINES 

intended to apply primarily to soft and clear but unfiltered river 
waters. Some waters will corrode cast-iron pipes much more 
rapidly than such a standard, while in other cases, especially for 
hard waters, the rate may be much slower. In all cases, therefore, 
careful judgment must be used. 

For riveted-steel pipe, when new, values of C may range about 
110 and upward. With age the value will drop to a range usually 
from 95 to 100, Hazen and Williams* state as a broad generalization, 
that steel pipe of a given size and age will carry the same quantity 
of water as cast-iron pipe of the same size and ten years older. With 
butt-strap circumferential joints, approximately flush riveting on 
the inside and special care to realize good hydraulic conditions, C 
for new pipe will rise to values about 120, falling with advancing age 
to values ranging from 100 to 110. 

For new welded-steel pipe with care at the joints to give, as 
nearly as may be, a smooth continuous surface, the value of C may 
be taken 120 to 130. With advancing age this will drop to values 
ranging from 100 to 110. 

Pipe of lead, brass, tin, glass, or of other like material giving a 
smooth semi-polished surface, will give values of C up to 140. 
A very slight roughening, almost imperceptible to the eye, will 
serve to reduce these values to 130 or 120 or less. 

For smooth-wood or wood-stave pipe, values of G from 120 to 
130 have been noted in several cases. 

For masonry or concrete conduits with smooth cement-plastered 
surface, values of G from 130 to 140 may be anticipated when new, 
falling with age and the development of a slime -covered surface to 
values ranging from 120 to 130, and to still lower values if accom- 
panied by flaking or scaling. Ultimate values of 120 or less will 
also be appropriate if the surfaces show slight waves or irregularities, 
as is often the case with ordinary contract work. 

For vitrified pipe a value of about 110 may be employed, thus 
allowing for some loss at each joint, due to roughness or a slight 
sudden expansion in cross -sectional area. 

In the preceding discussion regarding the values of the coefficient 
C, no specific reference has been made to the dependence of C upon 
size of pipe and hydraulic gradient or velocity. In so far, however, 
as the general indications of Kutter's formula are applicable to pipe 
flow, it appears that for a given assumed degree of roughness, the 
values of C increase with the hydraulic mean radius r and with the 
hydraulic gradient i or otherwise with the hydraulic mean radius 
and with the velocity. For large pipes and high velocities we may 
therefore anticipate relatively high values of C and for small pipes 
and low velocities, smaller values. 

The form of the Chezy formula, when compared with the known 
experimental facts regarding surface fluid resistance, shows, further - 

* loc. cit. 



GENERAL HYDRAULIC PRINCIPLES 11 

more, that we should anticipate such a dependence of G on the other 
factors, especially velocity or hydraulic gradient. 
Two practical questions then arise : 

1. For what sizes and hydraulic gradients are the values of C, as 
above indicated, intended to be applicable ? 

2. What is the general character of the change in the value of 
C with varying hydraulic mean radius or varying hydraulic 
gradient ? 

In answer to the first of these it may be stated that, broadly, the 
values given have been derived by observation from cases where 
for the most part the hydraulic gradient approximates -001, and the 
hydraulic mean radius ranges, say from -2(f) to 1(f). That is, the 
very small sizes of pipe are excluded from the range intended. As 
a middle range we may assume the values of C as stated, primarily 
applicable to cases approximating as follows : 

i=-00l 
r=-50(f) or 
D=2(i). 

We may then pass to the second query. Kutter's formula is, of 
course, intended to answer just this question, or more exactly to give 
C for any assumed value of the roughness coefficient n with any 
given value of i and r. 

We shall prefer, however, to take the direct results of pipe flow 
observations, as on the whole more satisfactory than the numerical 
values of the Kutter formula, which, as previously noted, was based 
primarily on the flow in open channels. 

A suitable analysis of the Williams and Hazen formula (see 
Sec. 4), and into the details of which we need not enter here, serves 
to show that if we denote by G the value of the coefficient G, 
which is properly applicable to a standard value of the hydraulic 
mean radius r and a standard value of the hydraulic gradient i , 
then the value G properly applicable to any value of r and i will 
be given approximately by the formula: 

0=0 ^(jf (16) 

That is, starting from standard values the value of C varies 
as the eighth root of r and the twenty- fifth root of i. It follows 
that the variation with i is much slower than with r. 

Assuming then any value of G as suited approximately to the 
values of r =-5 and i =-001, we may from (16) obtain an indica- 
tion of the suitable value of C for any other value of r and i, as 
desired. 

It should be here noted that the tabular values of C given by 
Williams and Hazen are stated to apply primarily to values r =l 
and i ='001. The law of variation is, however, the same, and the 



12 



HYDRAULICS OF PIPE LINES 



values r =-5 and i=-00l are here chosen as on the whole better 
suited to the general range of values of G above mentioned. 

It may also be noted that a suitable analysis of Kutter's formula 
shows likewise a very similar though not equally regular law of 
variation of G with r and i. 

Value o! G derived from / in Darcy's Formula. — A check on 
the value of G may be obtained by the use of the relation between 
the two coefficients of the Darcy and Chezy formulae. This relation 
has been noted in Sec. 5. In Table I corresponding values of / 
and G are given against diameter. It is seen that the table is not 
carried beyond a diameter of 2 feet. This is for the reason that 
this formula gives values too small for large pipe. The Darcy 
formula with the value of / proposed for clean pipe should not be 
employed for large pipes for this reason. It is readily seen that 
according to the formula showing the relation between / and C, the 
maximum value of G, no matter what the diameter, will be ^/4:00g= 
113*4. This value for large clean pipe is entirely too small, while 
on the other hand, for pipe up to perhaps 1 to 2 feet in diameter 
the values agree well with general experience. 

It may be noted also that Darcy recommends the value for / 
as stated in Sec. 5 to be applied to new clean pipe, with an increase 
in /up to 100 per cent for old and corroded pipe. This is evidently 
equivalent to a decrease in the value of G in the ratio 1-00 to 
1-41. For intermediate states of roughness or corrosion, inter- 
mediate values will naturally be taken. 







TABLE I 






d 


•001G6/d 


/ 


W 


G 


1 


•01660 


•03660 


7030 


84 


•2 


•00830 


•02830 


9092 


95 


•3 


•00553 


•02553 


10080 


100 


•4 


•00415 


•02415 


10650 


103 


•5 


•00332 


•02332 


11030 


105 


•6 


•00277 


•02277 


11300 


106 


•7 


•00237 


•02237 


11500 


107 


•8 


•00207 


•02207 


11660 


108 


•9 


•00184 


•02184 


11780 


109 


1-0 


•00166 


•02166 


11880 


109 


1-5 


•00111 


•02111 


12190 


110 


2-0 


•00083 


•02083 


12350 


111 



Hamilton Smith's Coefficients. — As the result of an extended 
examination of pipe flow data, Hamilton Smith has deduced a 
series of coefficients, varying with diameter and velocity. In the 
development of these coefficients a very large amount of data was 



GENERAL HYDRAULIC PRINCIPLES 13 

critically examined and great care was taken to eliminate doubtful 
results and to develop a set of values for the coefficient C, based on 
reliable and consistent observations. These values have, in conse- 
quence, been widely accepted as presumably the most reliable 
present expression of the results of actual experience. 

It should be noted that these coefficients do not contain allow- 
ance for varying degrees of roughness. It is understood that the 
observations on which they are based relate broadly to cast-iron 
and riveted-steel pipe with clean and smooth interior surfaces ; 
that is, to what may be taken as substantially new pipe. 

Smith's observations are reported in the original paper* in the 
form of tables giving values for C against varying values of diameter 
and velocity. In order to compare with other authorities giving C 
against diameter and hydraulic gradient, and for direct use when 
the hydraulic gradient is given rather than the velocity, it is con- 
venient to transform these values into an equivalent set giving 
C against diameter and hydraulic gradient. 

In Table II the coefficients are given in the original form, and 
in Table III the equivalent values are given in terms of diameter 
and hydraulic gradient. 



TABLE II 

Smith's Coefficients 

Values of C in Chezy Formula, for Clean Cast-Iron and 
Riveted-steel Pipe 

Velocities in feet per second 



Diameter 














r 










of pipe 


1 


2 


3 


4 


5 


6 


8 


10 


12 


16 


20 


Feet 
























•05 


— 


78 


82 


86 


88 


89 


91 


91 


91 


91 


— 


•1 


80 


89 


94 


97 


99 


101 


103 


105 


105 


105 


— 


1 


96 


104 


109 


112 


114 


116 


119 


121 


123 


124 


124 


1-5 


103 


111 


116 


119 


121 


123 


126 


129 


130 


132 


133 


2 


109 


116 


121 


124 


127 


128 


132 


135 


136 


138 


— 


2-5 


113 


120 


125 


128 


131 


133 


136 


137 


141 


143 


— 


3 


117 


124 


128 


132 


134 


136 


140 


143 


145 


147 


— 


3-5 


120 


127 


131 


135 


137 


139 


142 


146 


149 


151 


— 


4 


123 


130 


134 


137 


140 


142 


146 


150 


152 


153 


— 


5 


128 


134 


139 


142 


145 


147 


150 


155 


— 


— 


— 


6 


132 


138 


142 


146 


148 


154 


155 


— 


— 


— 


— 


7 


135 


141 


145 


148 


151 














8 


138 


143 


148 


151 


153 















* " The Flow of Water through Orifices, over Weirs and through Open 
Conduits and Pipes." J. Wiley and Sons, New York, 1886. Also "Trans. 
Am. Soc. C.E., 1883." 



14 HYDKAULICS OF PIPE LINES 

TABLE III 

Smith's Coefficients 

Values of C in Chezy Formula, for Clean Cast-iron and 
Riveted-steel Pipe 

These values are transformed from those of Table II in such manner 
as to make diameter and hydraulic gradient the determining 
variables. 



Hydraulic 




























gradient 












Diameter, 


feet 












parts in 




























1000 


•05 


•10 


10 


1-5 


20 


2-5 


30 


3-5 


4-0 


50 


6 


7-0 


8-0 


•2 










110 


116 


121 


124 


129 


135 


140 


144 


148 


•4 






96 


104 


114 


120 


125 


129 


133 


140 


144 


148 


152 


•6 






99 


108 


116 


122 


127 


131 


135 


142 


146 


151 




•8 






101 


110 


118 


124 


128 


133 


136 


144 


148 






10 






102 


112 


119 


125 


130 


135 


138 


145 


150 






20 






106 


116 


124 


130 


135 


139 


143 


150 








40 






111 


120 


128 


134 


140 


144 


149 










60 






113 


122 


130 


136 


142 


147 


151 










80 






114 


124 


132 


138 


144 


149 


152 










100 






116 


126 


134 


139 


145 


150 


153 










150 






118 


128 


136 


142 
















200 




89 


119 


129 


137 


















30 




92 


122 


131 




















40 




94 


123 


133 




















50 


78 


95 


124 






















100 


82 


99 
























200 


87 


102 
























300 


89 


104 
























400 


90 


105 
























500 


90 


105 
























1000-0 


91 


105 

























Between these two tables a value is readily selected for any 
combination of the variables within the range which they are 
intended to cover. 

Value o! C derived from Ship Resistance Experiments. — Ex- 
tended and refined investigations have determined to a high degree 
of accuracy the value of the coefficient of friction for the resistance 
of ships. This is usually expressed by a formula : 

R=kAv n . 
where R = resistance (p). 
A = area (f 2). 
v= velocity (knots or fs). 
n=an index usually taken at about 1-85. 
Jc= coefficient. 

Since the loss of head h is measured by the work done in carrying 
one pound against friction the length of the line (see Sec. 11), we 



GENERAL HYDRAULIC PRINCIPLES 



15 



readily derive an expression for h in terms of the above formula 
for resistance as follows : 

4 
Length of pipe occupied by 1 pound = — =^ 

4 
Wetted surface for 1 pound =—=, 

wD 



h=E (for 1 pound) X L 



4kLv n 
wD 



Since hydraulic radius r=D/4: this becomes 

kLv n 
wr 



(17) 



Comparing this with the Chezy formula it appears that if n=2 
we should have 



2 = 



Since, however, n=l-85, it is readily seen, in order that the 
Chezy formula with its index 2 may give the same value of h as 
(17) with index 1-85, we must put 



C 2 = 



wv 



15 



(18) 



Now abundant experiment has shown that for smooth iron and 
steel plates and taking v in (fs), the value of k may be taken as 
follows : 

Freshwater .... &=-0037 
Salt water .... &=-0038 

This gives values of C as follows : 



V 


c 


2 


. 137 


4 


. 144 


6 


. 149 


8 


. 152 


10 


154 



It will be noted that since w and k both vary directly with density, 
the value of C is independent of density. 

These values are undoubtedly applicable to large pipes with 
smooth surfaces. They furnish, moreover, a confirmation of values 
directly derived from pipe line observations, as noted previously. 
For small pipes where there is mutual interference between the 
filaments of flow and where the conditions between a flat plate and 
an indefinite body of water in relative motion cannot be realized, 
the resistance becomes greater and the value of C less, as previously 
indicated. Also where the surfaces are not smooth, due to corrosion, 
scaling or fouling, the coefficient of resistance will increase and C 
will decrease as before noted. 



16 HYDKAULICS OF PIPE LINES 

8. Total Friction Head in Pipe made up of Sections 
of Different Diameters 

Let A lt A 2 , A s , etc., denote respectively the cross sectional areas 
of the various sections of pipe. Let L l9 L 2 , L z , etc., denote the 
corresponding lengths. Let A 1 be taken as the reference area, and 
let A 1 =m 2 A 2 =m s A 3 , etc. 

Let v 1 =velocity in section of area A v 

Then m 2 v 1 = velocity in section of area A 2 . 

m 3 v 1 = velocity in section of area A z , etc. 
^=total friction head. 
Then from (3) we have 

h Lg rn^l n^l 
or^^V 2 (19) 



~(S> 



This expresses h in terms of a single velocity v x and the various 
ratios m with the other characteristics of the sections. 
Again from (13) we have similarly 



or 






For a circular cross section B—kit and r=Z)/4 and we have 



*-*£*(£*) 



C 2 D 5 



If the variation in diameter is moderate the coefficient C may be 
taken as constant at an average value and we have 

•"V41 to 



■) 



7T 2 C 2 \D 5 

Let D denote any standard or reference diameter and L the 
length of a conduit of this diameter which would have the same 
total value of h. Then 



9H- - 



m 



L Q =ZL %?) (22) 



Thus for illustration : 

Given L t = 500, ^=4. 
L 2 = 800,D 2 =3-5. 
£ 8 = 1000,2) 3 =3. 



GENERAL HYDRAULIC PRINCIPLES 



17 



What is the equivalent length reduced to a uniform diameter of 
3 feet. 

'3\ 5 

3 \ 



D 

D 2 

D_o 



=•2373 and 500 X -2373 = 118-7 
■4627 and 800 X -4627 = 370-2 



3\« 



land 1000x1 



= 1000-0 



£ n =1489-0 



Thus a uniform conduit, 3 feet in diameter and 1489 feet long, 
will have the same frictional loss as the actual conduit of varying 
diameters and 2300 feet long. 

In Table IV will be found a table of fifth powers for use in con- 
nection with formulse such as those of Sections 6 and 8. 





TABLE IV 




Fifth Powers of I 


•1 . 


•00001 




•2 . 


•00032 




•3 . 


•00243 




•4 . 


•01024 




•5 . 


•03125 




•6 . 


•07776 




•7 . 


•16807 




•8 . 


•32768 




•9 . 


•59049 




1-0 . 


1 -0000 




1-1 . 


1-6105 




1-2 . 


2-4883 




1-3 . 


3-7129 




1-4 . 


5-3782 




1-5 . 


7-5937 




1-6 . 


10-486 




1-7 . 


14-199 




1-8 . 


18-895 




1-9 . 


. 24-760 




2-0 . 


32-000 




2-1 . 


. 40-841 




2-2 . 


. 51-536 




2-3 . 


. 64-363 




2-4 . 


. 79-626 




2-5 . 


. 97-656 




2-6 . 


. 118-81 




H.P.L.— C 







7 . 


. 143-49 


8 . 


. 172-10 


9 . 


. 205-11 


. 


. 243-00 


1 . 


. 286-29 


2 . 


. 335-54 


3 . 


. 391-35 


4 . 


. 454-35 


5 . 


. 525-22 


6 . 


. 604-66 


7 . 


. 693-44 


8 . 


. 792-35 


9 . 


. 902-24 


. 


. 1024-0 


1 . 


. 1158-6 


2 . 


. 1306-9 


3 . 


. 1470-1 


4 . 


. 1649-2 


5 . 


. 1845-3 


6 . 


. 2059-6 


7 . 


. 2293-5 


8 . 


. 2548-0 


9 . 


. 2824-8 


. 


. 3125-0 


•1 . 


. 3450-3 


■2 . 


. 3802-0 



18 



HYDKAULICS OF PIPE LINES 



5-3 . 


. 4182-0 


7-7 


. 27068 


5-4 . 


. 4591-7 


7-8 


. 28872 


5-5 . 


. 5032-8 


7-9 


. 30771 


5-6 . 


. 5507-3 


8-0 


. 32768 


5-7 . 


. 6010-9 


8-1 


. 34868 


5-8 . 


. 6563-6 


8-2 


. 37074 


5-9 . 


. 7149-2 


8-3 


. 39390 


6-0 . 


. 7776-0 


8-4 


. 41821 


6-1 . 


8446-0 


8-5 


. 44371 


6-2 . 


. 9161-3 


8-6 


. 46043 


6-3 . 


, 9924-4 


8-7 


. 49842 


6-4 . 


. 10737 


8-8 


. 52773 


6-5 . 


11603 


8-9 


. 55841 


6-6 . 


12523 


9-0 


. 59049 


6-7 . 


13501 


9-1 


. 62403 


6-8 . 


14539 


9-2 


. 65908 


6-9 . 


15640 


9-3 , 


. 69569 


7-0 . 


16807 


9-4 


. 73390 


7-1 . 


18042 


9-5 . 


. 77378 


7-2 . 


19349 


9-6 


. 81537 


7-3 . 


20731 


9-7 


. 85873 


7-4 . 


22190 


9-8 


. 90392 


7-5 . 


23731 


9-9 . 


. 95099 


7-6 . 


25355 


10-0 . 


. 100000 



9. Minor Losses of Head 

Under this general head we include the various losses due to 
miscellaneous turbulence and deformation of stream line flow, as 
discussed in Sec. 1. 

(a) Loss o£ Head at Entrance. — When water passes from a 
reservoir into the open end of a pipe line, there is a loss of head 
depending on the velocity and on the rapidity of the acceleration 
from rest to normal velocity within the pipe. 

This is called the entrance loss and may be expressed by the 
formula : 



h-k v - 

Where v— velocity (fs). 

&= coefficient depending on form of entrance. 



(23) 



Elementary hydraulic theory with experimental observation 
serves to furnish approximate values of h as follows : 



End of pipe flush with reservoir (a) Fig. 1 
Pipe projecting into reservoir . (b) Fig. 1 
Conical or bell mouth . (c) (d) Fig. 1 



k 

50 
93 
•15 to 



04. 



GENERAL HYDRAULIC PRINCIPLES 



19 



As the value of v rarely exceeds 10 (fs) the value of h with an 
entrance as at (a) would not exceed -75 (f), while with a suitably 
tapering entrance as at c or d it may be readily reduced to -15 (f) or 
to an amount usually negligible. 




Fig. 1. — Forms of Inlet. 



(b) Loss Due to Abrupt Expansion in 
Size. — If the pipe line is made up of 
sections of varying diameters and the 
transition is sudden from one size to 
another, there will be corresponding 
sudden changes in velocity and re- 
sultant losses of head at these points of 
transition. 

For a sudden expansion (Fig. 2), 
elementary hydraulic theory furnishes 
the approximate formula : 








A 



Fig. 2. — Loss Due to 
Enlargement. 



K'-IJtKi-) * ™ 



Where A x and A 2 are the two areas 

v x and v 2 are the two velocities. 
Subscript 1 relates to the smaller size. 
Subscript 2 relates to the larger size. 

Putting this value of h in the form 



h-- 






or 






we have for the value of the coefficient : 



: =H:) 2 ° r (t- 1 ) 2 (25) 



according as h is related to v x or v 2 . 

(c) Loss Due to Abrupt Contraction in Size. — In such case 
elementary hydraulic theory shows that the loss is due primarily 




Fig, 3. — Loss Due to Contraction^ 



20 



HYDRAULICS OF PIPE LINES 



to the expansion beyond the contracted vein which is formed in the 
entrance to the smaller size ; see Fig. 3. The loss of head will 
therefore be determined by the relative areas A z and A v But the 
ratio A^\A X depends on the ratio A 1 /A 2 in a manner which experi- 
ment alone can determine. We may therefore express this loss by 
the formula : 



h-- 



k 2g 



(26) 



Where k has values, as in the following table, derived from 
experimental results by Weisbach. 



k . . 



10 -20 -30 
•362 -338 -308 



TABLE V 

•40 -50 -60 
•276 -221 -164 



70 -80 
105 -053 



■90 1-0 
015 



Regarding the same loss, Merriman* gives a formula for the 
contraction ration A 3 /A v Taking the nearest two significant 
figures this formula is 






•042 



VAJA 2 

Using this formula we find values as in Table VI for the coefficient 
k in (26). Note that in this formula the loss is referred to the 
velocity v v Fig. 3. 



TABLE VI 



^i/4| 



10 
333 



20 -30 -40 
306 -275 -243 



50 -60 
208 -168 



70 
123 



80 -90 1-0 
•076 -028 



Bellasisf quotes values for the contraction ratio AJAj^ which 
lead to the same values of k as quoted from Weisbach. 

(d) Loss of Head Due to an Obstruction. — In the case of a pipe 




Fig. 4. — Loss Due to Obstruction. 



with an obstruction, as in Fig. 4, it may be assumed that the loss 
of head is primarily due to the sudden expansion beyond the con- 

* "Treatise on Hydraulics." John Wiley and Sons, New York. Chapman 
and Hall, London. 

t "Hydraulics." Rivingtons, London. 



GENERAL HYDRAULIC PRINCIPLES 21 

tracted vein which is formed just beyond the reduced opening. 
Let subscripts 1, 2, 3 refer respectively to the reduced opening, to 
the full size and to the contracted vein, and as above let A denote 
area and v velocity. Then applying the formula of (b) for abrupt 
expansion we have for the loss due to the obstruction : 



»=(£-)*%" 



(27) 



The ratio A 2 /A 3 depends upon A 2 jA v Rankine gives the following 
empirical formula for the ratio A 3 /A 1 : 

A 3 _ -618 

4i"Vl— 618 (AJA^Y 
From which we readily find 

A 3 ^18 (28) 

Putting the above value of h in the form 

h =*% < 29 > 

we find by substitution in (27) and (28) the following values 
oik: 

TABLE VII 

AJA 2 -10 -20 -30 -40 -50 -60 -70 -80 -90 1-0 
k . .229 49 18 8-1 3-9 1-9 -86 -33 -07 

In connection with the same loss Bellasis* gives on experimental 
basis, values for the coefficient of contraction A 3 /A 1 as follows : 



TABLE VIII 

AJA 2 -10 -20 -30 -40 -50 -60 -70 -80 -90 1-0 
A 3 \A X -624 -632 -643 -659 -681 -712 -755 -813 -892 1-0 

These are the coefficients referred to in connection with the loss 
due to contraction, and differ but slightly from those given by 
Merriman's formula. The product of the two ratios (AJA^ {A 3 \A^) 
gives A 3 \A 2 . After the analogy of (25) the value of the coefficient 
k with reference to the lower velocity v 2 will be 



'=(£-')' 



(30) 



Using this formula with the values in Table VIII we find k as 
follows : 

* loc. ciu 



22 



HYDRAULICS OF PIPE LINES 



TABLE IX 

AJA 2 -10 -20 -30 -40 -50 -60 -70 -80 -90 1-0 
k . . 226 48 18 7-8 3-8 2-1 -80 -29 -06 

The agreement between the two sets of values in Tables VII and 
IX is very close. 

(e) Loss of Head Through Valves. — A valve may be considered 
as a special form of obstruction. The loss of head in flowing 
through partially open valves of the gate 
type, as in Fig. 5, or of the butterfly type, as 
in Fig. 6, is found to agree approximately with 
the values given by (27), (28), or Table VI. 
T ^ mi ^"s^ Therefore if we have given the area of opening 




& 



Fig. 5. — Loss Due to 
Gate Valve. 



Fig. 



6. — Loss Due to Butterfly 
Valve. 



and the area of pipe in the case of valves of these types, we may 
reach at least an approximate estimate of the loss of head by direct 
substitution in these formulae. 

Experiments on large gate valves made by Kuichling and J. W. 
Smith give results as below : 

TABLE X 



Ratio K/D 


•05 


•10 


•20 


•30 


•40 


•50 


•60 


•70 


•80 


(see Fig. 5) 




















Ratio A t /A % 


•05 


•10 


•23 


•36 


•48 


•60 


•71 


•81 


•89 


h for 24 ,, valve 


235 


100 


28 


11 


5-6 


3-2 


1-7 


•95 


— 



fcor 30" valve 333 111 23 9-4 5-2 3-1 1-9 1-13 -60 



If these values of k are compared with those of Table IX by 
plotting both sets on an axis of A 1 /A 2 , considerable divergence will 




Fig. 7. — Loss Dub to Plug Cock. 



GENERAL HYDRAULIC PRINCIPLES 23 

be noted. The agreement, however, is perhaps quite as close as 
could be expected, and the divergence in any ordinary case would 
be unimportant. 

In the case of a plug cock, as in Fig. 7, these values for the loss of 
head are relatively greater, as indicated in the following table : 



TABLE XI 

AJA 2 -10 -20 -30 -40 -50 -60 -70 -80 -90 1-0 
k . .430 93 35 15 6-8 3-1 1-5 -55 -11 

These various formulae and values for loss of head through 
valves rest largely on Weisbach's experimental results. These 
experiments were carried out for the most part on relatively small 
sizes and extension of these results to large sizes is attended with 
some measure of uncertainty. Further experimental results are 
much needed in relation to these various problems. 

(/) Loss o! Head due to Bends and Angles.— When bends, 
curves or angles occur in a pipe line, the change in the direction 
of flow will result in a loss of head depending primarily on the 
suddenness of the curvature or on the 

abruptness of deviation of the stream line t ' ^ -'J** 8 ^, 

flow. This loss is due to the work required hzzz zY^^%, 
to bring about the readjustment of the lines fr. -_- z : r ^ y% 
of stream flow and also to some degree of ^ >^S N v>\ 

abrupt contraction and expansion with V^^Vvl 

eddy formation as indicated in Fig. 8. fej ]\\ ' 

Experimental work during the past W;'l 

fifteen years in particular has served to «,//, 

call attention to two important features ill!', 

connected with the flow of water around a Jim 

bend or turn, as follows : * J sJ*i 

1. In the flow of water around a bend or Fm 8> _l ss at 
turn, the curve showing distribution of Elbow. 
velocity over the cross section suffers 

marked distortion. Instead of an approxi- 
mate ellipse the curve becomes distorted with the peak of maximum 
velocity carried over toward the outer or convex side of the curve. 

2. The influence due to a bend or turn extends far beyond the 
limits of the bend or turn itself. On the upstream side, at least 
for a little distance, the stream will show some changes in pressure 
and distortion of velocity curve due to the changes produced at the 
turn. On the downstream side, the influence of the elbow or turn 
extends to a very considerable distance, showing just below the 
turn marked distortion of the velocity curve and of the pressure 
distribution. This distance seems to vary with many factors, but 
is usually found between 50 and 100 diameters of the pipe. 

It follows that the actual influence due to a bend or turn is 



24 HYDKAULICS OF PIPE LINES 

distributed over a considerable length of pipe, and it is therefore 
necessary to define with care the meaning of the term " loss due to 
a bend or turn." As we shall here use this term, and with reference 
to the numerical values given, it will imply the difference between 
the total loss of head over the entire length of pipe influenced by 
the bend or turn, and the loss properly chargeable to an equal 
length of straight uniform pipe with the same hydraulic properties 
and operating with the same velocity. We may properly assume 
that this difference is due to the introduction of the bend with the 
resulting change in the direction of flow and the attendant circum- 
stances as noted above. 

Again the loss thus defined may be viewed in either of two 
ways : 

1. In the same manner as in (a), (b), (c) it may be related directly 
to the velocity head, v 2 /2g, through a coefficient k. In such case k 
will depend on the ratio of the radius of the pipe to the mean radius 
of the bend, on the angle covered by the bend, on the size of the 
pipe and doubtless on the velocity v. 

2. The loss may be expressed in terms of the length of straight 
pipe which, under like hydraulic conditions, would show the same 
loss. This gives a ready expression for the loss in terms of lineal 
feet of straight pipe in terms of diameters of straight pipe, or in 
terms of the loss due to a length of straight pipe equal to the 
mean radius length of the bend or turn itself. 

The earliest extended experiments on this subject were made by 
Weisbach with results as follows for the value of k in the formula : 

h^k— 

Let D= diameter of pipe. 

jR=radius of curvature. 
Then for an elbow or bend of 90° we have as follows : 











TABLE XII 










D/R 
RjD 

k 


•40 

2-50 

•14 


•60 

1-67 

•16 


•80 

1-25 

•21 


1-00 1-20 

1-00 -83 

•29 -44 


1-40 
•71 

•66 


1-60 
•63 

•98 


1-80 

•55 

1-41 


2-00 

•50 

1-98 



Likewise for varying angles of turn, presumably for D/R approxi- 
mately 1-6 we have values as follows : 

TABLE XIII 

Angle . 20° 40° 50° 80° 90° 100° 110° 120° 
k . . -05 -14 -36 -74 -98 1-26 1-56 1-86 

Weisbach's experiments were made with relatively small pipe, 
and for which the Darcy coefficient may be assumed to vary from 



GENERAL HYDRAULIC PRINCIPLES 25 

•025 to -030. Recomputing the values in Table XII into terms 
of the equivalent length of straight pipe, we have the following : 









TABLE XIV 








D/R . 


•40 


•60 


•80 1-00 1-20 1-40 


1-60 


1-80 


2-00 


R/D . 


2-50 


1-67 


1-25 1-00 -83 -71 


•63 


•55 


•50 


L x . 


•78 


•89 


1-17 1-61 2-44 3-67 


5-45 


7-83 


11-00 


L t . 


• 1-87 


2-13 


2-80 3-87 5-87 8-80 


1307 


18-80 


26-40 



In the above table L x gives the lengths of straight 2 -inch pipe 
with Darcy coefficient =-030 equivalent to the excess loss due to 
the elbow, while L 2 gives the similar lengths for 4-inch pipe with 
Darcy coefficient =-025. 

Williams, Hubbell and Fenkell,* reporting on an extended 
investigation on curve loss in 90° elbows in large pipes, give results 
for a 30 -inch pipe which indicate a continuously increasing loss 
with increase in the ratio R/D. The general results for this size 
pipe are shown in the following : 







TABLE XV 








R/D 


. 24 


16 


10 


6 


4 


2-4 


k . 


. 1-4 


•96 


•82 


•65 


•27 


•24 


Loss 


. 180 


123 


105 


83 


35 


31 



In this table the second line gives the value of k in the formula 
h=kv 2 /2g, while the third line gives the length in feet of straight 
30 -inch pipe equivalent to the excess loss due to elbows. 

The velocities involved in the experiments leading to the results 
in Table XV were all low — from 1 to 3-5 feet per second. 

As a result of an extended investigation on 90° elbows in 3 -inch 
and 4-inch pipe, Brightmoref found results which he expressed 
in the form of curves showing values of the excess loss in terms of 
head. The value of G in the Chezy formula for average pipe of 
these sizes may be taken at 110. Using this value and converting 
the values drawn from Brightmore's curves we have the following : 





TABLE XVI 






Size of Pipe 4 inches. Velocity in (fs). 




R/D 


5 7-5 


10 


10 


•24 3-7 -20 31 -21 


3-3 


8 


•26 4-0 -27 4-2 -28 


4-4 


6 


•30 4-7 -32 4-9 -33 


51 


4 


•21 3-4 -29 4-5 -28 


4-4 


2 


•34 5-4] -38 6-0 -39 


6-1 





112 17-5 1-22 190 1-18 


18-4 



Values of k and of equivalent length. 

* " Transactions Am. Soc. C.E., 1902." 

t " Proc. Inst, ag.," Vol CLXIX, 1906-7. 



26 



HYDRAULICS OF PIPE LINES 



In the above table the first column under each velocity gives the 
value of k in the equation h=kv 2 /2g, while the second column 
gives the length in feet of straight 4-inch pipe equivalent to the 
excess loss due to elbows. 







TABLE XVII 






Size of Pipe 


3 inches. 


Velocity in (fs). 




R/D 


5 


7»5 


10 


14 


•13 1-5 


•14 


1-7 -13 


1-5 


12 


•17 2-0 


•13 


1-6 -12 


1-5 


10 


•24 2-8 


•19 


2-2 -19 


2-3 


8 


•30 3-5 


•33 


3-8 -34 


4-0 


6 


•34 4-0 


•38 


4-5 -39 


4-6 


4 


•30 3-5 


•32 


3-7 -30 


3-5 


2 


•43 50 


•42 


4-9 -42 


4-9 





1-12 13-1 


1-31 


15-2 1-18 


13-9 



Values of k and of equivalent length. 

In the above table the first column under each velocity gives 
the value of k in the equation h=kv 2 /2g, while the second column 
gives the length in feet of straight 3 -inch pipe equivalent to 
the excess loss due to elbows. 

In Brightmore's experiments the elbow of ratio R/D—0 was 
represented by a Tee plugged at one outlet. 

Reporting on the results of an extended investigation on 90° 
elbows in a 6 -inch pipe line, Schoder* gives results as follows : 









TABLE XVIII 












Velocity in 


(fs). 






\\D 




3 


F 






10 


16 


20 


•34 


9-4 


•27 


7-8 


•19 


5-6 


•14 4-3 


15 


•15 


4-2 


•09 


2-6 


•05 


1-4 


•016 0-5 


10 


•21 


6-0 


•16 


4-5 


•11 


3-3 


•08 2-5 


8 


•28 


7-8 


•21 


6-2 


•17 


51 


•13 41 


6 


•28 


7-8 


•21 


6-1 


•17 


51 


•14 4-3 


5 


•14 


4-0 


•12 


3-5 


•11 


3-3 


•11 3-3 


4 


•24 


6-6 


•18 


5-3 


•16 


4-7 


•12 3-8 


3 


•21 


5-8 


•18 


5-1 


•16 


4-7 


•12 3-8 


2-16 


•22 


6-2 


•19 


5-4 


•17 


5-1 


•13 4-1 


1-90 


•25 


7-0 


•21 


6-2 


•19 


5-8 


•16 4-9 


1-76 


•24 


6-8 


•24 


6-9 


•23 


6-8 


•29 8-8 


1-34 


•39 


10-8 


•33 


9-6 


•30 


8-9 


•26 8-1 



Values of k and of equivalent length. 
* " Tra,»s ? Am, Spc, C.E.," Vol. LXII. 



GENERAL HYDRAULIC PRINCIPLES 27 

In the above table the first column under each velocity gives the 
value of k in the equation h=kv 2 /2g, while the second column gives 
the length in feet of straight 6-inch pipe equivalent to the excess 
loss due to elbows. 

In converting from one set of values to the other the following 
values of the coefficient G are employed, being those given by 
Schoder for the particular pipe and velocities employed. 



V 


G 


3 


120 


5 


122 


10 


124 


16 


126 



In connection with the same investigation Schoder reports 
measurements on certain lengths of 8 -inch pipe line containing 
slight bends, one of 3-8°, and a reverse curve of 2*18° in one direction 
and 2-81° in the other, all of which failed to indicate any measurable 
loss due to the bends. 

In discussing Schoder 's results, Davis* gives values of the loss 
due to 90° elbows in a 2 -inch line for a number of different values 
of E/D. These are given in terms of loss in head in feet. From 
these results we may derive the following : 

TABLE XIX 

Size of Pipe 2 inches. Velocity in (fs). 
R/D 5 15 



10 


•31 


2-4 


•43 


3-4 


5 


•19 


1-5 


•40 


31 


2-5 


•26 


20 


•40 


31 


115 


•46 


3-6 


•60 


4-7 


•OOf 


1-65 


12-9 


1-60 


12-5 



Values of k and of equivalent length. 

In the above table the first column under each velocity gives 
the value of k in the equation h=kv 2 /2g, while the second column 
gives the length in feet of straight 2 -inch pipe equivalent to the 
excess loss due to elbows. In converting from one set of values to 
the other a typical value of (7=110 was employed. 

Experiments on a wide variety of small and medium sized elbows 
and fittings have given indications as below for the value of k in 
the equation h=kv 2 /2g. 

* loc. cit. 
f Represented by a Tee plugged at one outlet. 



28 



HYDKAULICS OF PIPE LINES 



TAB! 


.E XX 








Description 


Velocities 




k 




§" Black mall, elbow (old) 


2, 


5, 10 


•82 


•76 


•72 


§" Galvan. mall, elbow (new) 


2, 


5, 10 


•57 


•53 


•50 


1" Black mall, elbow (old) 


2, 


5, 10 


•76 


•70 


•67 


1" Cast-iron elbow (old) 


2, 


5, 10 


1-02 


•95 


•90 


2" Mall, iron elbow . 


2, 


5, 10 


•74 


•72 


•69 


3" Cast-iron elbow . . 


5, 


10, 25 


•54 


•54 


•53 


4" Cast-iron elbow . 


5, 


10, 25 


•61 


•58 


•54 


6" Cast-iron elbow 


5, 


10, 25 


•50 


•48 


•48 


2" Cast-iron Tee and plug (water 












leaving branch) . 


2, 


5, 10 


1-85 


1-91 


1-88 


Same Tee as above (water en- 












tering branch) . 


2, 


5, 10 


1-43 


1-55 


1-63 


3" Cast-iron Tee and plug (water 












entering branch) 


5, 


10,25 


1-45 


1-43 


1-37 


4" Cast-iron Tee and plug (water 












entering branch) 


5, 


10, 25 


1-55 


1-41 


1-22 


Same Tee as above (water leaving 












branch) .... 


5, 


10, 25 


1-24 


1-17 


1-10 


4" Cast-iron Tee filled in to make 












square elbow 


5, 


10, 25 


M0 


1-07 


1-07 



Throughout these various values of the loss due to 90° elbows, 
there is found considerable divergence among the various experi- 
ments and abundant evidence of irregularity and inconsistency in 
many of the results. These are doubtless due, for the most part, to 
the difficulty of eliminating the influence of obscure and secondary 
causes which have no representation in the scheme of the investiga- 
tion or in the final formulae deduced. The trend of the investiga- 
tions indicates, as would be expected, a general decrease in the 
loss with increasing value of the radius of the bend. The results 
of Brightmore and Schoder all agree, however, in indicating an 
increasing value for the loss, to a local maximum, for values of 
B/D from 6 to 8 or 10, and followed by diminishing values for 
greater values of B/D. The results of Davis indicate a similar 
condition, but were not carried far enough to determine the de- 
creasing values of the loss for values of B/D greater than 10. The 
lowest values of the loss for values of B/D less than 6 or 8 seems 
to be indicated for values from 2 to 4. Schoder's low values for 
B/D=15 and high values for B/D=20 seem to be abnormal and 
inconsistent with each other and with the other results. The results 
of Williams, Hubbell and Fenkell stand alone in indicating a 
continuously increasing value of the loss for values of B/D from 
2-4 to 24. Either some undetected source of error is involved in 
these results, or the value of the loss is gradually increasing toward 
a maximum at some value of B/D beyond 24, after which it will 
decrease. It is obvious that as the ratio BjD indefinitely increases, 



GENERAL HYDRAULIC PRINCIPLES 29 

the excess loss, as compared with an equal length of straight pipe, 
must approach zero and hence the values of the loss cannot indefi- 
nitely increase with increase of R/D. 

These irregularities and inconsistencies indicate two things : 

1. The experimental evidence at hand is neither sufficient in 
amount nor consistent enough in character to permit the develop- 
ment of a satisfactory general formula for curve loss. 

2. The influence of secondary features, such as irregularities in 
form or surface condition, or irregular and turbulent conditions 
of flow which occasionally develop, seem sufficient in importance 
to entirely mask the general trend of the system of phenomena 
expressed in terms of velocity, size and value of RjD alone. 

General Considerations. — The following general considerations 
may be noted in connection with these various minor losses of head, 

The loss due to an abrupt enlargement is usually more serious 
than that due to an abrupt contraction. In either case, as noted, 
the loss is due primarily to the eddy formation resulting from an 
abrupt enlargement. When the ratio A 1 jA 2 is small and the loss 
is correspondingly large, the loss for abrupt expansion will be the 
larger of the two. When the ratio A X \A 2 approaches 1 and the loss 
itself becomes relatively insignificant, the loss for contraction 
will become the larger of the two. These results are readily verified 
from the formulae and tables of (b) and (c). 

The flow of a liquid past a sharp edge is always a fruitful source 
of eddy formation and of loss of head. The presence of a sharp 
edge projecting into a flowing stream means an abrupt change in 
the cross -sectional area and a corresponding abrupt enlargement 
or contraction or both. 

In order to minimize these various sources of loss it is clear that 
all abrupt changes in stream line flow must be avoided, and in 
particular any abrupt enlargement of cross -sectional area, any 
flow over or across sharp edges, or any abrupt change in the general 
direction of flow. 

10. General Resume of Loss of Head in Pipe 
Line Flow 

It thus appears, in connection with the flow of water through 
a pipe, that there are a considerable number of sources of loss of 
head. These may be classified as follows : 

1. Loss of head due to friction denoted by h as discussed in 

Sees. 2-5. 

2. Loss of head due to miscellaneous turbulence caused by 

abrupt changes in direction or velocity as discussed in Sec. 9 
and denoted also by h. These various items of loss due 
to turbulence are all expressed in the general form 

*-£ 

2? 



30 HYDKAULICS OF PIPE LINES 

where k is a coefficient to be determined according to the 
details of the case, and as discussed in Sec. 9. 
It appears that the values comprising loss (2) are usually small 
in comparison with those of the frictional loss (1), and with suit- 
able dispositions they may in many cases be rendered negligible. 
In any case taking the values comprising loss (1) proportional to 
v 2 , the total loss of head may be expressed in the form 

-£+»£ < 31 > 

Where Eh denotes the sum of all the coefficients k for the various 
secondary losses discussed in Sec. 9. This may be put in the form 

H^+f> 2 (32) 

In this equation Hk/2g may be considered as a supplementary 
term modifying the principal term L/C 2 r. In most cases, as noted 
above, the supplementary term will be small compared with the 
principal term, and in consequence it is very often neglected ; 
or otherwise is considered as included within the margin of un- 
certainty which must attach to the selection of a value of C, or 
of the roughness coefficient in Kutter's formula from which C is 
determined. 

In dealing with practical problems it is therefore customary to 
select from judgment either C direct or a value of the roughness 
coefficient which will determine C, and to assume that within the 
necessary margin of uncertainty concerning these values, the 
resulting value of h or friction loss, will adequately represent all 
loss of head throughout the pipe. 

It should not be forgotten, however, that the items comprising 
loss (2) may require special recognition, and in such case suitable 
estimates or allowances must be made. Thus in case the flow 
passes through a partially closed valve the loss of head may be very 
considerable, and in such case due allowance must be made for the 
value of such loss independent of the estimates for determining the 
value of the friction loss by the usual formulae. 



11. Special Relations 

The following relations which will be found of frequent value in 
the discussion of various problems may be conveniently assembled 
at this point. 

(a) Friction head &=loss of energy per pound of water in 

traversing distance L. 

Lv 2 

(b) -^-=loss of energy as in (a)=work done in carrying one 

r pound of water through distance L against 

frictional resistance in pipe. 



GENERAL HYDRAULIC PRINCIPLES 31 

v 2 

(c) -^-=frictional or skin resistance per pound of water in pipe. 

(d) it^42y=weight of water in pipe in pounds. 

(e) — -^ — =wAh— total frictional or skin resistance for pipe as a 

6 r whole. 

(/) wAv=v&te, of flow (pounds per second). 
(g) l/w^4v=time for flow of one pound past a given section. 

W "772 — = ra te of work against friction for entire pipe. 

(*) — t^ — X — 7-=77H-=^=work done against friction in entire 
v C 2 r wAv C 2 r ° 

pipe in the time required for the flow of one 

pound past a given section. 

WLv 2 

(j) —^2 — = Wh=worh done against friction in entire pipe in the 

time required for the flow of weight W past a 
given section. 

(h) — ^ =total kinetic energy in pipe. 

(I) — =time rate of change of total kinetic energy. 

g at 

. . wALv dv 1 L dv . , . .. . . .. 

(m) -r-X — 7- = — i-== change m kinetic energy in the 

v ' g dt wAv g dt & &J 

time required for the flow of one pound past a 
given section. 

(n) — ^-=measure of an accelerating head. 

Suppose an unbalanced pressure p at one end 
of the pipe, acting on the cross sectional area A. 
Then pA=total pressure acting as an acceler- 
ating force on contents of pipe wAL. Then 

. dv Force pAg , L dv p 

acceleration =— =— — = £ —rF and =-=£■— 

dt Mass wAL g dt w 

accelerating head. That is — -=- is the measure 

of an accelerating head acting over the cross 
section A and producing an acceleration dvjdt : 
or again, an unbalanced head H acting at one 
end of a pipe will produce an acceleration in the 
velocity of movement of the contents of the pipe 
measured by dvjdt— gHjL. 

(o) —= change in kinetic energy in the time required for 

& the flow of weight W past a given section. 



32 



HYDRAULICS OF PIPE LINES 



12. Distribution of Velocity Over Cross Section 

of Pipe 

Pitot tube measurements have been made by many investigators 
over the cross section of a pipe running full. The results are not 
altogether concordant and the law of distribution of velocity is by 
no means definitely assured. Errors of observation, effects due to 
pulsating movements and to turbulence, and obscure influences due 
to velocity and roughness have doubtless contributed to this result. 
In Fig. 9 let the velocities at varying distances along the radius be 
observed and set off from AB parallel to the line of flow and located 



A C F 


£ 




—-^ 


L 


M 




> } 


L 




^ 


> 



B EG 

Fig. 9. — Variable Flow over Cross Section of Pipe. 



at the points of observation. The general result will be a curve of 
the form CDE showing a velocity AC at the outer surface of the 
stream, a velocity XD at the centre and a varying distribution of 
velocity between the centre and the surface as shown by the curve. 
The chief points of interest in such a curve are the following : 

1. The relation of A C to XD, the velocity at the surface to that 

at the centre. 

2. The relation of the mean velocity to XD, the velocity at the 

centre. 

3. The radius at which the actual velocity will equal the mean. 

In Fig. 9 let AF be the mean velocity. Then the actual velocity 
KL at the radius XK will equal the mean velocity AF. 

Bazin, Williams, Hubbell and Fenkell, Cole, Schoder and others 
have made observations on the form of such a curve and the 
observations generally indicate for CDE a curve shaped much like 
an ellipse. There is considerable variance in the relation between 
AC and XD. Williams, Hubbell and Fenkell came to the conclu- 
sion that the surface velocity was about -50 that at the centre. 
Cole's measurements indicated a ratio of -60 and more. It is clear 
that the mean velocity will be the height of a cylinder AFGB whose 
volume is equal to the solid of revolution formed by revolving A CD 
about XD. If the elliptical character of CDE is assumed, the latter 
will comprise a cylinder plus a half ellipsoid of revolution. It is 



GENERAL HYDRAULIC PRINCIPLES 33 

well known in geometry that the mean height of a half sphere or 
ellipsoid of revolution equals 2/3 the height of the body. Hence 

AF=AC+2/3MD. 
Denoting AF by v m , AG by v s and XD by v c , we find immediately 

or v m =2/3v c +l/3v s . 
If then v s = • 50 v c we have 
v m =-833v c . 
If v s =-60v c we have 

In any case it is readily seen from the properties of the ellipse, 
that the point where the height of the curve is 2/3 the maximum 
height is at *745r. Hence if the ellipse may be assumed as a close 
approximation to the general character of the curve of velocity 
CDE, the mean velocity will be found at a point close about *75r 
from the centre. 

In any measurement of importance dependence should not be 
placed on such a relation and one or preferably two complete 
traverses with the Pitot tube should be made across diameters at 
right angles to each other. 

Bellasis has compiled from a number of sources the values given 
in Table XXI showing the ratio v m jv c for a number of different kinds 
and sizes of pipe, and at various values of v m . 

TABLE XXI 

Table Showing Values of Ratio of Mean Velocity to 
Centre Line Velocity in Pipe Line Flow* 

Diameter Mean velocities in feet per second 
Kind of pipe of pipe 

in inches 78 1-5 2-5 3-5 5 8 14 

Brass seamless . . 2-70 -73 -77 -79 -80 — — 

Cast iron . . . 7-5 — — -80 -81 -82 -83 -84 

Cast iron . . . 9-5 — -80 -81 -82 -83 -84 -85 

Cast iron with deposit 9-5 — -81 -81 -82 -82 -83 -83 

XT . . , .., ( 12 — -83 -83 -84 -85 -85 -85 

New iron coated with I 16 _ . g2 . g3 . g4 <g5 _ _ 

coal-tar . • ^ 30 .75 . 8 3 -84 -85 — — — 

Cement ... . 31-5 — — — -85 -86 — — 
New iron coated with 

coal-tar . . 42 — — — -86 — — — 



13. Mean Velocity and Mean Velocity Head Over 
Cross Section of Pipe 

The experimental investigation of the velocity over the cross 
section of a pipe gives a result as indicated in Sec. 12. It becomes 
a problem of interest to determine, from such a distribution of 

* Bellasis, "Hydraulics," p. 129. Rivingtons, London. 
h.p.l. — D 



34 



HYDRAULICS OF PIPE LINES 



velocity, the volume flow and kinetic energy of the stream, or 
otherwise the mean velocity and the mean velocity head. 

In Fig. 10 let the annular rings denote a series of elements into 
which the cross -sectional area may be divided, and over any one of 
which the velocity may be considered uniform. 

Let a denote the area of any one such annular ring or element, 
v the velocity over this element, A V, the corresponding element of 
volume flow and l\K that of kinetic energy. 

Then &V=av 
wav 3 



AK: 



29 

And V= Zav 
K=^-Lav 3 

where, as usual, 2 denotes the summation of a series of terms, all 
similar in character and made up by multiplying each a by its 
appropriate value of v in one case and v 3 in the other. 

The general procedure may there- 
fore be outlined in the following 
steps : 

1. By means of Pitot tube obser- 
vations obtain a series of values 
of v at a series of points along the 
diameter of the pipe, or preferably 
along two diameters at right angles. 
These points may be so chosen as 
to come in the middle of a series 
of annular rings, as in Fig. 10. 
These values for the two diameters 
may then be averaged so as to 
give a mean series for the mid- 
points along the radius. 

2. The values of a are then 
determined in accordance with the 

successive values of|the radius. \ 

3. Each value of a is multiplied by the corresponding value of v 
and the products summed. The result will give V, the total volume 
flowlin (f3s). This|divided by the cross sectional area of the pipe 
will Jive the mean v for the entire area, 

4. Each value of a is multiplied by the cube of the corresponding 
value of v and the results summed. This sum multiplied by w/2g 
will give the kinetic energy passing the given section in one second 
of time. This divided by wV, the weight passing the section per 
second, will give the mean kinetic energy per pound, or otherwise 
the mean velocity head. 

This general method is of such wide application that it will be of 




Fig. 10. — Annular Elements of 
Cross Section. 



GENERAL HYDRAULIC PRINCIPLES 



35 



interest to develop a slightly different method for treating the Pitot 
tube observations. 

In terms of the calculus, and assuming the width of the annular 
rings indefinitely small, we have 

Area of annular element 2-nrdr. Then 
dV =2-nvrdr 

dK— — v 3 rdr. 

g 

In Fig. 11 let ABC denote the values of v plotted along the 
radius OX. Then at a series of points along the radius take the 





Fig. 11. — Integration for 
Volume Flow. 



Fig. 12. — Integration for 
Energy Flow. 



value of v, multiply it by r and set off the product as the ordinate 
of a new curve OPQR. Thus DP=ODxDE and similarly for all 
other points. Then the curve OPQR will represent the distribution 
of the product vr along the radius. The area of this curve taken by 
planimeter, or otherwise, will give the integration of vrdr, and this 
multiplied by 2 it will give the value of V. 

Similarly in Fig. 12 let ABC denote the values of v 3 plotted along 
OX and OPQR, a curve derived as in Fig. 11 by multiplying each 
value of the ordinate as DE by the corresponding radius OD. Then 
the curve OPQR will represent the distribution of the product v z r 
along the radius and the area of this curve by planimeter or other- 
wise will give the integration of v B rdr, and this multiplied by nwjg 
will give the value of K. 



14. Hydraulic Gradient 

Writing again the general energy equation as in (1) we have 

f+g+*+fc=iS„ (33) 



36 



HYDKAULICS OF PIPE LINES 



In Fig. 13 let R denote the reservoir with water level at NN. Let 
ACDB denote a pipe line with discharge at B. Let XX denote the 
base or datum line from which we measure gravity head z. Then 
within the reservoir at the surface NN we have p/w=b (where 6= 
atmospheric or barometric head), v—0, h=0, z=H and hence 
b/w-{-H =Z . That is the entire head Z is measured by the 
gravity head H plus the atmospheric head b/w, and the gravity 
head H is measured by the elevation of the surface NN 
above the datum line XX. To simplify the present discussion 
suppose the pipe ACDB of uniform size. Then the value of v will be 
uniform throughout the length, as also the value of the velocity 
head v 2 /2g. Suppose this value laid off as a vertical distance from 





T 


u 


W 










J 






N 


N 


M 







=: 


_= 


<r~ — 




G 




ti 




R 










F*- 




S 










A 


SS:!a! ^5a5a ::: C 




















He 
















• 




P 






X 










^1) 


B 












i 


E 







Fig. 13. — Hydraulic Grade. 



NM downward giving the line QR. Then the vertical intercept 
between NM and QR will, at any and all points, give the value 
of v 2 /2g. 

Suppose likewise a line TW laid off above NM at a vertical 
distance equal to the atmospheric head b/w. 

Next suppose the value of h to be computed for the various points 
along the line, giving in each case the total loss of head from the 
entrance A to the given point P, and let the values be laid off as 
vertical ordinates from QR downward, giving the line QS. It may 
be noted that strictly speaking the line QS will start at Q a little 
below the velocity head line QR, the distance between the two lines 
at Q denoting the value of h at Q which will be measured by the 
entrance loss as discussed in Sec. 9. 

Next it is clear that the line of the pipe ACDB gives graphically 
at any point the value z as the vertical distance above the datum 
XX. Hence at any point P it follows that z=PE, h=GF, v 2 /2g= 



GENERAL HYDRAULIC PRINCIPLES 



37 



JG, b/w=JU and Z —EU. Hence comparing with equation (33) it 
follows that we must have pjw—FP-\-JU. 

That is, the total pressure head within the pipe is measured by 
the sum of the two intercepts FP+JU. But the latter of these 
measures the atmospheric head or pressure and hence the former 
must measure the head or pressure above the atmosphere ; or in 
other words, the head or pressure which would be indicated by an 
ordinary pressure gauge. 

It follows that if an open-end vertical tube were inserted in the 
pipe at P the water would rise in such tube up to the level F., thus 
indicating directly the pressure head at P above the atmosphere. 
Again, if a series of such tubes were inserted along the length of the 
pipe from A to D the water levels would all lie on the line QS. 

The line QS thus determined, is called the hydraulic gradient or 
hydraulic grade line. Again any point on this line as F is called 
" hydraulic grade " for the corresponding point P on the pipe line. 
The hydraulic grade line may thus be denned as a line any point of 
which is vertically above the pipe line a distance equal to the 
pressure head above the atmosphere in the pipe line at such point ; 
or conversely as a line any point of which is vertically below the 
static level line a distance equal to the sum of the velocity head and 
lost head for the corresponding point in the pipe line. 

It must not be forgotten that the pressures and pressure heads 
indicated by the hydraulic grade line are measured above the 
atmosphere and that in terms of absolute pressure the pressure in 
the pipe or the pressure head will be greater than the amount thus 
indicated by the pressure or pressure head due to the atmosphere. 
Thus at P (Fig. 13), the pressure head above the atmosphere is 
denoted by PF while the absolute pressure head is denoted by 
PF+JU. 

Again, in Fig. 14 let ACB denote a pipe line with entrance at A 
and discharge at B. Let QFS be the hydraulic gradient and TMW 




Fig. 14. — Hydraulic Grade. 



the line of atmospheric head above QS. Then at the points D and 
E the pressure in the line will just balance the atmosphere. From 
A to D and from E to B the pressure will be greater than the 
atmosphere, while between D and E the pressure will be less than the 



38 HYDRAULICS OF PIPE LINES 

atmosphere. At G the pressure head will be a minimum, measured 
by GF below the atmosphere or CM above absolute pressure datum. 
It is thus clear that if any part of the pipe line crosses or rises above 
the hydraulic grade line the pressure within such part of the line 
will be less than that of the atmosphere. 

If, furthermore, the line should rise sufficiently above the 
hydraulic grade line QS to just reach the atmospheric head line TW, 
then the absolute pressure at such point in the pipe will be zero ; or 
more exactly it will be reduced to the pressure of water vapour under 
the temperature of the water at the given point. The pressure of 
water vapour at ordinary temperatures is very small and it is 
customary to neglect its influence on ordinary problems of pipe line 
flow. The water at such point would therefore be in the same 
physical condition as in a bell jar exhausted under an air pump to a 
perfect vacuum, or more exactly to a vacuum corresponding to the 
vapour pressure at the temperature of the water. 

If again the line of the pipe should at any point rise above the 
line TW it would imply a negative pressure in the water, or in 
other words a tension instead of a compression. A tension cannot, 
however, be developed in a stream of water. In answer to the 
conditions which might tend to set up a tension the stream will 
break and the conditions of continuous steady flow will no longer 
obtain. 

Actually, turbulence, unsteadiness and interruption of the 
conditions of steady flow will result somewhat before the pressure 
is reduced to zero. This is due in large measure to the liberation, 
under the reduced pressure, of the air which is normally held in 
solution in the water. 

The rise of the pipe at any point above the hydraulic grade line 
results in a condition of unsteady flow with irregular turbulence and 
a tendency to set up pounding or water hammer, especially in the 
part beyond the point which rises above the grade line. Since the 
grade line varies with the velocity or with the rate of discharge, it 
is clear that in any pipe line, such as in Fig. 14, there will be some 
critical velocity which will just bring the grade line to the angle C. 
For lower velocities G will lie below the grade line andffor higher 
velocities above it. 

In any such case the trouble resulting from a point C lying above 
the grade line may be avoided by a suitable reduction of the velocity 
of flow with the consequent rise in the grade line. 

15. General Formula for Capacity 

Taking the Chezy formula for velocity we have 
7=^=rate of flow (f3s). 



GENERAL HYDRAULIC PRINCIPLES 39 

Then 7= — r- or F 2 — (34) 

P* P 

For a round pipe running full this becomes 

F= y_^! = .3927CK*2)« (35) 

o 

Where Z>=diam. in feet : or solving for i we have 

6-484F 2 ,_, 

In Table IV will be found the fifth power of numbers which with 
an ordinary table of squares and square roots will serve to readily 
solve numerical problems involving these formulae. 



16. Carrying Capacity of Round Pipe running 
Partly Full 

In Fig. 15 let AB denote the surface of the water. Then for the 

area ABC we have 

D 2 
A=-£ (0— sin cos 0), 

and for the wetted perimeter ACB we have 

P=DB. 

Then referring to (34), the general formula for capacity, we find 

by substitution and change of form : 

V 2 (0-sin cos 0) 3 

— \yi) 



CHD** 



64 



Table XXII gives for 5° intervals the 
values of the right-hand side of this equa- 
tion, and also of z/D. By means of this 
table we may readily determine the rela- 
tions between V, C, i, D and or z. Thus 
if the pipe, the slope and the quantity are 
given and z is required, we substitute the 
given values in the left-hand side of (37) 
and find the numerical value. Then from 
the table we find approximately the value 
of and then the value of z. Again if z or 
0, G, i and V are given, we readily find D 
by taking from the table the value of the 
right-hand member of (37) and solving the 
resulting equation for D. 

It will be noted that the capacity is a maximum for approxi- 
mately 155° and z=«95Z). More exact analysis by means of the 
usual calculus treatment for maxima and minima shows the capacity 




Fig. 15. — Flow in Cir- 
cular Section Par- 
tially Filled. 



40 



HYDRAULICS OF PIPE LINES 



a maximum for 0=154° 05', z/D=-9496. It thus appears that a 
round pipe running with an open segment at the top about 5 per cent 
of the diameter in height, will have a capacity nearly 10 per cent more 
than if running full. This result arises from the fact that for a 
reduction in section up to this point the beneficial influence due to 
the relatively rapid decrease in the wetted perimeter is more 
influential on V than the relatively slow decrease in the area of the 







TABLE XXII 






e 


a 


b 


e 


a 


b 





•0000 


— 


95 


•5436 


1-602 


5 


•0019 


— 


100 


•5868 


2-015 


10 


•0076 


— 


105 


•6294 


2-464 


15 


•0171 


— 


110 


•6710 


2-932 


20 


•0301 


— 


115 


•7113 


3-401 


25 


•0469 


— 


120 


•7500 


3-854 


30 


•0670 


•0007 


125 


•7868 


4-272 


35 


•9040 


•0023 


130 


•8214 


4-639 


40 


•1170 


•0062 


135 


•8536 


4-944 


45 


•1465 


•0148 


140 


•8830 


5-178 


50 


•1786 


•0315 


145 


•9096 


5-334 


55 


•2132 


•061 


150 


•9330 


5-424 


60 


•2500 


•111 


155 


•9532 


5-443 


65 


•2887 


•187 


160 


•9698 


5-406 


70 


•3290 


•299 


165 


•9830 


5-324 


75 


•3706 


•454 


170 


•9924 


5-207 


80 


•4132 


•659 


175 


•9981 


5-073 


85 


•4564 


•918 


180 


1-0000 


4-935 


90 


•5000 


1-233 








a= 


ratio of z to D, depth t 


o diameter, 






b = 


value of (0- 


-sin cos 0) 


3 in equation (37), 





64 

section. Beyond this point, however, the influence due to the de- 
crease of area becomes paramount and the capacity decreases for 
a further reduction in area. 

It should be noted that these results presuppose perfectly steady 
conditions and a smooth water surface without waves. Actually 
there will be some wave formation, especially if there is any depar- 
ture from uniform motion, and under such circumstances the 
theoretical gain for a partially filled cross section will be much 
reduced. Under ordinary conditions it can hardly be considered 
desirable to attempt to run a round pipe partially full for the sake 
of an increase in capacity. The same remarks apply generally to 
all forms of cross section, though with certain tunnel forms the 
relative advantage might be somewhat greater than with round 
pipe. 



GENEKAL HYDKAULIC PRINCIPLES 



41 



17. General Problem of Steady Flow 

Let AB (Fig. 16) denote a pipe line of varying cross section 
terminating in a nozzle at the discharge end B. 

Let ^=loss of head due to friction and turbulence in general. 

Starting from the upper or entrance end of the pipe line A, the 
total loss h will have continuously varying values, increasing along 
the length of the pipe according to the circumstances affecting skin 
friction and turbulence. 

Let a, A v A 2 , A z , etc., denote respectively the cross section 
areas of the nozzle and of the various lengths of pipe of which the 
line is composed, counting from the discharge end back. 




Fig. 16. — General Problem of Pipe Line Flow. 



Let a=m 1 A 1 —m 2 A 2 =m 3 A 3 > etc (38) 

Where m v m 2 , m 3 , etc., form a series of coefficients relating the 
various areas A v A 2 , etc., to a. 

Also let u, v v v 2 , v 3 , etc. = velocities in the nozzle and in the 
successive sections of area A v A 2 , etc. 

Then v 1 =m 1 u i v 2 =m 2 u, etc (39) 

Let H= gravity or static head relative to the base line XX. 
In— total lost head from A to any point P. 
?/ 1 =head due to absolute pressure (usually atmospheric) on 

surface of water at source. 
2/ 2 =head due to absolute pressure at point of discharge. 
b = atmospheric head at any point P. 
p = pressure above atmosphere at any point P. 
z=gravity head at any point P—PE. 
Usually y lt b and y 2 are all the same. The heads y x and y 2 in 
the general case, however, may have any value. Thus we might 
have compressed air acting on the surface NN, or again air under 
a reduced pressure or partial vacuum. Likewise the head y 2 implies 
any pressure whatever at the point of discharge, such for example 
as that due to discharge into a tank with compressed air, or into 



42 HYDRAULICS OF PIPE LINES 

a partial vacuum, or again discharge under water and therefore 
against an absolute pressure due to the water plus the atmosphere. 

Usually the differences in atmospheric pressure are negligible, 
and in such case, and where no other pressures are involved, the 
terms in b and y may be omitted from the equations. 

In the general case we have, as in (1), the Bernouilli equation for 
steady flow between A and P as follows : 

(H+ yi -h)=Z+b+ J+z (40) 

or -Jfc^+6+g-H,) (41) 

Putting (41) into words we have : 

The difference between the values of the total lost head h corre- 
sponding to any two points such as P is equal, with the opposite 
algebraic sign, to the aggregate change in the actual head (pfw-\- 
b-\-v 2 /2g-\-z) between the same two points. 

In applying these equations there are five variable terms in- 
volved, of which four must be known in order to determine the 
fifth. In the usual procedure h is expressed as a function of v by 
means of the formulae of Sees. 2-6 and 9. 

At the point of discharge through a nozzle or valve, however, 
there is usually a special loss which must be otherwise expressed. 

Taking the datum for z at the level of the valve or nozzle, the 
absolute pressure head just back of the valve will have the value 

5+*"*+"-^-* < 42 > 

where h is the total lost head between A and B. 

Just beyond the nozzle, at the point of discharge, the absolute 
pressure head is y^. 

The difference between these two measures the net or resultant 
pressure head acting between the two sides of the valve or through 
the nozzle. This net pressure head is then H-^-y 1 —y 2 —v 1 2 l2g—h. 
Adding to this the velocity head v-^j2g we shall have the total net 
head available for producing the discharge velocity u at the mouth 
of the nozzle and hence the external velocity head u 2 /2g. This 
total net head is then H J r y 1 —y 2 —h. 

We may then consider this total net head at the valve as trans- 
formed through the valve or nozzle into the velocity head u 2 j2g. 
This transformation is accompanied by loss, and hence we may write 

w 2 

~=AH+ yi - yi -h] (43) 

where / is the efficiency of transformation. 

We shall usually call / the nozzle coefficient or coefficient of dis- 
charge. The value of / is always less than 1, approaching 1 as an 
upper limit. The difference (1—/) denotes the fraction of the total 
net head lost by friction and turbulence in the valve or nozzle. 



GENEKAL HYDRAULIC PRINCIPLES 43 

Putting Ay for y 1 —y 2 and substituting for h its value as in (19) 
we have 



*-/[*+4,-r(3£)..] (44) 



2<? 



(ti+ Ay) 



whence u=\/ 1 /m 2 L\ (45) 

Note that Ay will be +, — or according as y 1 is greater than, 
less than or equal to y 2 > 

Knowing u we may then find any other velocity from (39). 
The known values of h, z and v for any point in the line may then 
be substituted in (40), serving to determine p/w or p for such point. 
Thus the entire hydraulic conditions throughout the length of the 
line become known. 

If we may neglect the loss through the nozzle, (45) becomes 



/ H+Ay 
u=V I +2 ,/*^A (46) 



2g' T \C*r) 

If in addition we have 2/i=2/ 2 > (46) becomes 

/ * 
«=v i r/ m2L \ ( 47 ) 

If, furthermore, the pipe is of uniform diameter throughout we 
have 



/ H 



u=\ \_ m*L (48) 

2g^ C*r 

The value of u for any other combination of the determining 
characteristics is readily derived from the general form in (45). 
We have, also, by combining (42) and (43), 

g=/(f +5 -^) (49) 

™-I= 2 fr 6+ ^-T (50) 

>-HC(r-v) : m 

If the loss through the nozzle is negligible, we have 

|+6-2/ 2 =^a-«i 2 ) (52) 



44 



HYDRAULICS OF PIPE LINES 



Equation (50) gives the gauge pressure just back of the valve, 
while (51) and (52) give the drop in pressure through the valve. 

If, furthermore, y 2 =b, as is the case with atmospheric discharge, 
we have ^ „2 

iT^ 1 — «■> ( 53 > 



18. Pipe Line Operating with a Free 
Surface Flow 

In certain cases a pipe line may be operated partially full, thus 
developing all the hydraulic characteristics of open channel flow 
(see Fig. 17). 




Fig. 17. — Flow in Open Channel. 



Neglecting any changes in atmospheric head and taking p as 
the gauge pressure, we have for the general equation, as in (1) : 

w 2g 

At all points in the line sensibly near the surface of the stream 
we shall have p=0, and hence as the general equation of flow near 
the surface : 

At A just inside the pipe, and where the velocity v has been 
developed, we shall have 



H—h A =2 q + z A 



or z 



=*-(£W) 



(54) 
(55) 



This means that just inside the entrance the level of the water in 
the pipe will have dropped below that in the reservoir a distance 
(v 2 /2g-{-h A ), representing the head due to the velocity v plus some 
entrance loss h A) as discussed in Sec. 9. 



GENERAL HYDRAULIC PRINCIPLES 45 

At the discharge end B we have 2=0, and hence 

H ~^b=~ (56) 

Subtracting (56) from (54), we have 

h B -h A =z A (57) 

Expressing this in words, the loss of head in the line between A 
and B is the difference in level between these same points. This 
means that whatever the actual gradient on which AB is laid, the 
velocity will automatically take such a value as will consume the 
entire head H in doing three things, as follows : 

(1) Producing the velocity v (vel. head v 2 /2g). 

(2) Overcoming loss at entrance A (lost head k A ). 

(3) Overcoming losses in line (lost head h B — h A =z A ). 

Or again, in a more restricted way we may say that no matter on 
what gradient the pipe is laid, the velocity will automatically take 
such a value as will consume in lost head the entire difference in 
level between B the point of discharge and A a point just inside the 
inlet and where the velocity v has been set up. 

For the loss in head between A and B we have as in (3) : 
, , Lv 2 



^A—Q^r-ZA 



or from (54) : 



As in (23) we may express h A in the form kv 2 /2g where k must be 
determined by estimate. Substituting this value in (58) and 
solving for v we have 



/— 



^ L (59) 



2g ' C 2 r 

It is very common to omit (l+&)/2g in comparison with L/C 2 r, 
or otherwise to assume that its value is absorbed in the uncertainty 
regarding the value of C. In such case we have approximately : 

Lv 2 
H=£ r (60) 

A 2 C2rH 

and v l — — j — 

But H/L=i the gradient ; whence 

v=C-\/ir, as in equation (2). 

In case the gradient is very steep and the developed velocity 
very high, as where the pipe is used as a free surface spillway, the 
velocity head will be too great to permit of omission from the 
equation. In this case equation (59) must be employed. This 



46 HYDRAULICS OF PIPE LINES 

equation will, moreover, not be exact, since there will be, at the 
upper end of the pipe, a certain length over which the water will be 
accelerating from some low velocity at the very point of inlet up to 
the full velocity v. The effective length of the line for velocity v, so 
far as friction loss is concerned, is therefore not L but something 
less than L* Moreover, throughout this part of the pipe, the cross 
sectional area of the stream will gradually contract with increasing 
velocity, and with corresponding change in r. Otherwise one may 
say that, of the total length L, a part will operate under the ulti- 
mate velocity v and a fixed r, while the remainder, at the upper end, 
will operate under a variable v and a variable r from the inlet to the 
point where the conditions are practically steady. These con- 
ditions indeed prevail in any case, but with a low gradient and low 
velocity both the velocity head and influence on the friction head 
due to the accelerating length may safely be neglected. With high 
velocities we may still, usually without serious error, neglect the 
influence on h due to the accelerating length and thus obtain from 
(59) a satisfactory value by taking £=length of the line and r 
constant* 

It should also be noted that both (59) and (60) admit of direct 
solution in case the value of r is given. On the other hand, in case 
the quantity or volume flow V is given, the equations are implicit. 
The hydraulic mean radius is not known in advance, since it will 
depend on v. Such a case will call for a trial and error procedure. 
A value of r is assumed, the value of v found and the resulting F. 
Thus by successive trial a value of r will be found which will give the 
V desired. The relations of Table XXII will conveniently aid in 
handling problems of this character. 

In order to illustrate the error involved in using for a case of free 
surface flow with high velocity, equation (2), Sec. 2, instead of (59), 
the following example may be noted. 

Let angle of slope =45°. 
Then »=-7071. 
Let £=100 f. 
(7=120. 
r=-5. 
Then #=70-7. 

Then taking k=0, substituting in (59) and solving we find v=49. 

If, however, we neglect the velocity head and use equation (2) we 
shall find v=71-4. 

The proper value of G to use in the case of such very high gradients 
and velocities is subject to very grave uncertainty. As has been 
noted in Sec. 7, there is reason to anticipate increasing values of C 
with increasing velocities . Actual experimental values are , however, 
lacking. If we attempt to deduce any indication from the results of 
ship resistance experiments, we find an increase approximately as the 
seventh root of the speed (see Sec. 7). If then a value of 110 were 
taken as appropriate for any given case with i;=10 we should have 



GENERAL HYDRAULIC PRINCIPLES 



47 



corresponding to v=50, a value of (7=138. If this value were used in 
(59) the result would be v=52-l, while in equation (2) we should 
have v=82. There is need of further experimental data regarding 
the values of C appropriate to such velocities, and also regarding the 
importance of the length at the upper end affected by acceleration. 



19. Pipe Line Connecting Two Reservoirs 

Certain interesting problems arise in connection with the use of a 
pipe line connecting two reservoirs with the water at different 
levels. Such problems may present themselves under three 
principal cases. 

Case 1. Pipe running full or under pressure. See Fig. 18. Let B 
and S denote the two reservoirs with water levels maintained 




Ho 


-H, 


~M 


M 






— 


~p H J~ s~ 





Fig. 18. — Flow in Conduit between Connecting Reservoirs. 



steadily at NN and MM, and let AB denote the connecting pipe 
with nozzle or valve at B. Let XX, at the level of the point of 
discharge B, denote the datum for measuring gravity head z, and 
let H 1 =the excess head MB. 

Then this case is covered by thefgeneral treatment of Sec. 17. 
Omitting any influence due to difference of atmospheric pressure 
between the locations of the two reservoirs, we have for /\y in 
equation (45), the value —H v 

If in addition we may take /=!, we have 



Vt^ 



>-#! 



or for a single uniform diameter of pipe 



* + * 



(61) 



u 



Vf* 



-Hi 



2g^C 2 r 



(62) 



Comparing these equations with (47) and (48) it appears, as we 
might expect, that in this case the velocity of discharge is the same 
as that for discharge into the air with a total head=-M G, the differ- 
ence in level between the two reservoirs. 



48 



HYDRAULICS OF PIPE LINES 



Again the head just at the point of issue is 



w 



At some point P in the reservoir, on the same level as B but 
where the velocity has become negligible, we shall have 

H P =H V 

This implies during the operation of bringing the stream to rest 
between B and P, the loss of head u 2 /2g as a direct result of the 
transfer of the energy of translation into the energy of turbulence, 
and its ultimate dissipation as heat. 

Case 2. Pipe running partly full or with free surface. See Fig. 17. 
In this case the formulae of Sec. 18 apply directly and no further 
discussion of the case is required. 

Case 3. Pipe running partly full at upper end and full at lower 
end. See Fig. 19. This case will arise when the lower reservoir is 




Fig. 19. 



-Flow in Line Partly with Free Surface and Partly 
with Full Section. 



maintained with its level above the discharge end of the pipe while 
the upper reservoir is maintained with its level only partially cover- 
ing the inlet end ; or otherwise when the water admitted at the 
inlet end is less than the capacity of the pipe under full flow. 

Under these conditions the upper end of the line will operate with 
free surface flow, while toward the lower end it will operate full and 
under pressure flow. 
Let F=volume flow. 

-4 1 =section of stream between A and C. 
^4 2 =full section of pipe. 
Assume a valve or nozzle fitted at B serving to partially close the 
outlet into the lower reservoir. 

Let a=area of opening. This may have any value from to 

the full area A 2 - 
Let %= velocity between A and C= Y \A V 
v= velocity in full pipe= V/A 2 . 
u~ velocity through opening. 
/= coefficient of discharge. 
x = distance BG. 
z— elevation CD. 



GENERAL HYDRAULIC PRINCIPLES 49 

We may then consider the level G as equivalent to the level of an 
open surface reservoir with the pipe operating under the hydraulic 
conditions of Fig. 18. Referring to the general treatment of Sec. 17, 
and applying equation (44) to the present case, we have 

£-'(-**-!*) < 63 > 

With V known, v becomes known. If then x is known or fixed, 
all the terms in (63) become known except u. The equation is then 
solved for u from which the area a is found. This gives, therefore, 
the area of opening which, with a given V will serve to maintain G 
at any desired point. 

If, on the other hand, V and a are fixed then u becomes known and 
also v. If then the pipe BG is straight we may put z=x sin and 
solve (63) for x finding 

u 2 

2gf +Hl 
*= — — r- (64) 

sm 6 ~~c*? 

This gives, therefore, the location of G for a given V with a fixed 
value of a. If the line BC comprises sections of varying diameters, 
then equation (63) will take the form 

H*-M^)-S) (65) 

where the term Z(Lv 2 jC 2 r) includes all the complete sections 
between B and C and x is measured from the junction next below C. 
The location of the section containing C must be made by inspec- 
tion or trial. In each of these sections, with V fixed the value of v 
is known. Then with a fixed and u known, all terms in (65) become 
known except x and z. If the length containing C is straight we 
may put 

z=z 1 -\-xsin 0. 

Where z x is the elevation of the junction point next below C and 6 
is the angle of slope of the length containing C. The equation is 
then solved for x. 

If the length containing the point C is curved instead of straight, 
then z must be expressed in terms of x according to the geometry of 
the line, or otherwise the equation is readily solved by trial and 
error. A point C is assumed and the values of z and x are found and 
tested by substitution in equation (65). 

If, on the other hand, the point G is fixed, then the right-hand side 
of (65) is completely known and it only becomes necessary to solve 
for u and then to find a. The values of v x and A x in the free surface 
section, AG (Fig. 18) are readily found by equation (37) or Table 
XXII of Sec. 16. 

H.P.L.— E 



50 HYDKAULICS OF PIPE LINES 



20. Power Delivered at Discharge End of Line 

Let V— volume rate of flow (f3s). 
w= density (pf3). 
P= power (fps). 
H = static head (above atmosphere). 

H=net he&d=H —h. 
P=(H -h)Vw. 
Then assuming discharge into the air and neglecting any differ- 
ence in atmospheric pressure between the entrance and discharge 
ends of the line, we have 
64ZF* 
° n 2 C 2 D 5 

^P=HVw=(H V-%fei)v, (66) 

In equation (66) it is seen that with given V and other conditions, 
the value of P is readily found. Conversely, however, if P is given 
and V required, the equation becomes a cubic in V and will usually 
be most conveniently solved by approximate or trial and error 
methods. 

To determine the conditions for a maximum value of P we 
differentiate with reference to V, place the result —0 and solve for 
H thus finding : 

192LV* 

or h=H /S 

Interpreted in words this means that the maximum power will be 
delivered at the discharge end of the pipe line when the lost head h 
amounts to one-third the total head H . 

For this set of conditions we readily derive the following expres- 
sions for the values of the velocity v, the volume flow V and the 
maximum power P r 



m' 



2V 3£ 



(67) 



V=*™M. (68) 

Pn= ™CHjD> 

12V3£ 

These formulae presuppose, of course, that the water called for in 
(68) is available. With a fixed flow of water the maximum power 
will, of course, be developed with the minimum lost head h. 



GENEKAL HYDEAULIC PRINCIPLES 51 



21. Piping Systems 

Several important and interesting problems arise in connection 
with the flow of water through a ramifying system, as in Fig. 20. In 
such a system each run of pipe from one junction to the next forms 
a unit or element. These elements are shown by the numbers in 
the diagram. In any such system it is readily shown that the 
number of elements is equal to the number of outlets plus the 
number of junctions. Also that the number of junctions is one less 
than the number of outlets. Hence if there are n outlets there will 
be (2n~ 1) elements in the system. The characteristics required to 
determine each element are the diameter and velocity. If neither 

F 




6^ 

Fig. 20. — Branching System. 

of these is known then the problem.will comprise 2 (2n—l) variables 
and we must have an equal number of equations in order to find a 
unique solution. If either the diameter or velocity is known, the 
number of variables is reduced to (2n—l). 

In connection with such a system of piping three typical problems 
may arise. 

1. Given a definite system with fixed dimensions of pipes, with 
given valve openings at the discharge ends and with given 
coefficients of discharge, all under a fixed head H. The valve 
openings may be anything from to full opening. Required the 
total discharge and the discharge at each opening. 

2. Given a series of locations relative to the reservoir or source of 
head (pump for example) with a stated discharge at each point. Let 
the head at the supply point (pump or reservoir) be fixed and 
constant. Required a system of piping suited to accomplish these 
ends. 

3. Same as (2) but without a fixed or given head at the source. 



52 HYDRAULICS OF PIPE LINES 

Required a system of piping together with a head suited to accom- 
plish the stated deliveries. 

We shall only refer briefly to the principal features of these 
problems. 

Problem 1. — In this problem everything is given regarding the 
various elements of the system except the values of v. We shall 
therefore, with n outlets, have (2w— 1) unknown velocities and shall 
require (2n— 1) equation for their determination. We obtain these 
as follows : 

(a) One equation for each outlet, or n in all, by tracing the loss 

in head from source to discharge and equating the head at 
the point of discharge to the original head diminished by 
the loss. 

(b) One equation for each junction, or (ft— 1) in all, by equating 

the flow in the pipe leading to the junction to the sum of 

the flows in those leading from it. 
This will furnish a system of (2n—l) simultaneous equations, the 
solution of which will give the values of v, one for each element of 
the system. Naturally the actual solution becomes rapidly burden- 
some with increase in the number of elements, but the principle 
remains the same and for the problem as stated there seems no way 
of evading the details of the solution in this manner. 

The development of such a system of equations will be sufficiently 
illustrated by writing those for a simple Y branch with unequal 
legs (see Fig. 21). We shall assume that each element is formed of 
pipe of uniform diameter. If this is not so the principles discussed 




Fig. 21. — Single Y Branch. 

in Sec. 8 readily furnish a means for making provision for this 
feature. Let H in general denote the difference in external head 
between the supply point A and any other point in the line. If the 
flow is from an elevated reservoir at A, H=the difference in level 
between A and the given point. If the head at A is supplied by a 
pump, H is the difference between the total head furnished by the 
pump and the elevation at the given point, both reckoned from the 
same datum, the pump for example. 

Let m 2 , m 3 denote respectively, as in Sec. 17, the ratio between 
the discharge orifice area and the area of the pipe, for branches 2 
and 3, and / 2 , / 3 the corresponding coefficients of discharge. Then 



GENERAL HYDRAULIC PRINCIPLES 



53 



using in general the notation of Sec. 17 we have, for the values of 
the head at the discharge points C and D : 

V _, Y„ £iV -W 



2gm, 



\ H '' 



L lVl * 






(70) 



2gm 3 * "L" 8 OAi <V) 
Also, « 1 4 1 =v 2 4 2 +«) 3 4 3 .. 



(71) 
(72) 



These three equations will serve to determine v l9 v 2 , v 3 , and 
hence the flow condition becomes completely known. From (70) 
and (71) we derive 



# 2 -V 



2s/ a w s 



2 '2 



:#, 



-»4» + ^}- (73) 



This gives a relation between v 2 and v 3 , and shows that such rela- 
tion is entirely determined by the conditions affecting those two 
lines. Also if either v 2 or v 5 is known or given, the other may be 
determined from this relation. 

The same relation may be generalized to apply to any of the 
branches of a Y connection, such as 2 and 3 of Fig. 20. To this end 
we may rewrite (73) as follows : 

1 f *0-fa-&W r^+^rl <"> 



(b,- 



2g ' O^r 



y=( fl --s)-w.] 



Where H 2 and H 3 = differences in external head along ABC 
and ABD respectively, p c and p d = pressures at C and D and / 
and m are both taken=l. This shows that if we know the pressure 
heads at the ends of the two legs of any Y branch with the other 
controlling conditions and the velocity in one of them, we may 
find that in the other. 

Problem 2. — In this case we know the discharges but neither 
the velocities nor the sizes of the pipes. There will be, therefore, for 
each pipe two unknowns, a size and a velocity, and hence in all 
2(2n— 1) unknowns. We may assume that the discharge openings 
are fixed and the coefficients of discharge known or assumed. From 
the known discharges we know the product vA for each terminal 
element of the system. From these and equations of the form 
v 1 A 1 =v 2 A 2 -\-v 3 A s we may work back and find the numerical value 
of vA for each element of the system. This will give us (2n— 1) 
equations. 

We can then write n equations of the form (70), but shall still 
lack (n— 1), or as many equations as there are junctions. It results 
that the system is indeterminate. There is no unique solution 
and the conditions may be fulfilled in an indefinite variety of ways. 
A physical analysis of the conditions in a single Y branch, for 
example, will readily lead to the same conclusion. 



54 HYDKAULICS OF PIPE LINES 

It follows that in order to make the problem definite we must 
fix (n—l) of the variables. These must furthermore be fixed in 
a manner consistent with the other equations. We have already 
seen from (74) that the velocities in the two legs of any one Y 
branch are not independent, and that their relation is determined 
by the conditions in these legs. We are not, therefore, free to fix 
arbitrarily either velocity or area in both legs of any one Y branch. 
We may, however, fix one, and as there are (n— 1) such Y branches 
this will represent the number of variables to be arbitrarily assumed. 

We therefore proceed by fixing arbitrarily the area or velocity 
in one leg of each Y branch (for example 7, Fig. 20). The velocity 
or area in the same leg will follow conversely from the relation 
vA = V, while the velocity in the other leg 4 will follow from (73) 
or (74). In using (74) we shall need the pressure heads at points 
such as B and C. These are readily found by working back from 
the discharge ends. Thus knowing the heads at the discharge 
points F, G, we readily work back, from the determined conditions 
in 4 and 7, to the head required at C and hence to the pressure 
head at C. Having found the velocity in 4 we find the area from 
vA — V, and thus by working back from the discharge points all 
conditions become known up to the point B. Then from the head 
necessary at B and the known head at A the conditions in element 1 
are readily found, and thus the entire system. 

As a variant on the above procedure for this problem we may, 
for a given leg such as 7, assume the hydraulic gradient i instead 
of either the area or velocity. The diameter then follows from 
(35) and thence the velocity. The procedure otherwise is in general 
the same. 

It may easily result that, in working back in this way, an im- 
possible set of conditions will develop. Thus we find that the head 
required at B is greater than that available at A . This will be due 
to the arbitrary assumption made regarding size or velocity or 
hydraulic gradient. In such case the assumption must be varied 
until the system as determined becomes feasible and consistent in 
its various parts. 

In order to avoid this, the procedure may be advantageously 
varied as follows : 

In Fig. 22 let ABC and ABB represent the profiles along the 
lines of a single Y branch. Let N be the pressure elevation at A, 
that is A N— pressure head supplied by pump at initial end of AB. 
Then from the fixed discharges, discharge openings and coefficients 
of discharge at D and C, the pressure heads at D and C just back 
of the valve are readily found (see (50)), Let these be set up as 
CE and DF. Then EG and FH are available for friction heads 
along the fines ABC and ABB respectively. We may then draw 
any line, straight or broken, from E to N and take such fine as giving 
the hydraulic gradient along the pipe line ABC. The value or 
values of i thus determined may then be used, as before, in (35) 



GENERAL HYDRAULIC PRINCIPLES 



55 



to find the diameter, and thence otherwise as before. The same 
general method is readily applied to a more complex system. The 
procedure is, of course, entirely similar in case the point A of Fig. 22 
is fed from an elevated reservoir. In such case, however, the point 
N (Fig. 22) should, strictly speaking, represent the head just inside 
the pipe 1 at A rather than the level of water in the reservoir. The 
difference is usually negligible, or otherwise we may assume the 
conditions in element 1, thus fixing the elevation of N and the 
hydraulic gradient as far as the first junction B, and then determine 
the remainder as above. Other variations in detail will readily 
occur to the engineer having to deal with such problems. The 
main purpose of the arbitrary assumptions which are made must 
be to produce, if necessary, by trial and error, a final system, 
harmonious and consistent in its various parts. 

Problem 3. — In this case we do not have a fixed head at the 
initial point A. We are therefore free to either fix arbitrarily or 
tentatively the head at A and proceed as in the first method dis- 




Fig. 22. — Branching System. 



cussed under Problem 2, or otherwise, and preferably to fix, accord- 
ing to judgment, the gradients along some one line from the dis- 
charge end back to A and thus determine the head at A and the 
various sizes along the line. The head thus determined must then 
be accepted as applying to the other fines, or if not acceptable 
for them, another trial must be made. Thus by trial and adjust- 
ment an acceptable system may ultimately be determined. 

Certain further interesting problems in connection with a rami- 
fying system arise when a reservoir head and a pumping head are 
both applied at different points in the system, as in Fig. 23. This 
is frequently met with in connection with municipal distributing 
systems. 

Problem 4. — Assume the system of Fig. 23 to be entirely given 
with all elevations of outlets, sizes of pipes ; valve openings and 
coefficients of discharge. Assume further at any given time a 
given level N in the reservoir. Assume a head AM maintained by 
the pump. Required the flow throughout the system and the 
movement of the surface N. 

This problem falls directly under the treatment of Problem 1. 



56 HYDRAULICS OF PIPE LINES 

We have in the arrangement of the diagram seven elements in the 
piping, and we shall therefore require seven equations. There are 
three outlets, E, F, G, one outlet K under the head represented 
by the level N, and three junctions. These will furnish the needed 
seven equations, and we may then find the flow in all elements, 
including the line 5 leading to the reservoir. The flow here will, 
however, raise the level N in accordance with the dimensions of 
the reservoir, and as the level rises the flow in 5 will become less 
and less. It follows that a complete study of the case would require 
a series of solutions of the equations for a series of varying eleva- 
tions of N taken at suitable intervals, and corresponding to intervals 
of time which will result directly from the surface area of the 
reservoir and the mean rate of flow along 5 for any one interval. 
In this way a time history of the rise of level N could be developed. 
Ultimately a point would be reached where the flow in 5 will become 
zero, the level N will remain stationary and the flow in the remainder 
of the system will become steady at the values determined for this 
condition. This terminal elevation of N may be directly found by 
assuming immediately the condition of zero velocity in 5, and hence 
the practical elimination of this branch with the reservoir from 
the problem. We then proceed with the remainder of the system 
exactly as in Problem 1, thus finding the flow along all elements 
of the system. With these known the pressure head at any given 
point is readily found. We therefore find the pressure head at the 
point C, and this must give therefore the elevation of the surface 
N above C. 

In case the head AM maintained at A is, at the start, below 
the level required to maintain N stationary, then the flow in 5 
will be reversed, the reservoir will become a feeder, and the level 
N will fall until it reaches a point where the conditions will be 
steady with N stationary. In any case with a fixed level M the 
level of N for steady conditions is readily found by assuming 5 
with the reservoir removed from the system and then proceeding as 
above noted. Conversely, if N is fixed, and it is desired to know 
the level at which M must be maintained to keep N stationary, 
we should have a system with 5 pipe elements (assuming element 
5 removed) and with an unknown head AM. This will, in the 
given case, make six unknowns. We have three outlets, two 
junctions and a given pressure head to be realized at the point C. 
These will furnish the necessary six equations. 

Problem 5. — In this case we assume given the discharge at the 
various outlets, a given head AM and a level N to be maintained 
stationary. Required sizes suited to realize these ends. 

The level N being stationary, the pipe 5 is inoperative, and we 
may therefore assume it removed so far as matters of velocity and 
flow are concerned. We shall then have as unknowns, five velocities 
and five areas. We shall be able to write five equations of the form 
vA — V, two for the junctions and one for the given pressure head at 



GENERAL HYDRAULIC PRINCIPLES 



57 



the point G. We must, therefore, arbitrarily fix two of the un- 
knowns, as, for example, the velocities in 6 and 7, and then proceed 
as in Problem 2. 

Or otherwise, and preferably, we may set up at the outlet, just 
back of the valve, vertical lines representing the pressure heads 
required by the discharges (see EP, FR and GQ, Fig. 23). We may 
then draw from P, Q and R hydraulic gradient lines back to M, 
which must, however, fulfil the following conditions : 

1. The gradients from P and Q must have a common point on 

the vertical through D, as at U. 

2. The gradient from U to M must pass through 8. 

3. The gradient from R to M must meet that from U to M on 

the vertical through B, as at T> and the two must be coin- 
cident from T to M. 

The gradients thus determined will then serve to determine 
the sizes of the elements of the system, as in Problem 2. 




Fig. 23. — Branching System with Double Supply. 



In case the pump shuts down, the system becomes reduced to 
that of a single reservoir source, and may be examined as in Problems 
1, 2 and 3. 

Various other combinations may present themselves in connection 
with problems involving ramifying systems, especially such as 
that of Fig. 23. In all cases, however, they may be investigated 
by the general methods outlined in the present section. 

In the case of reversed flow through such a ramifying system, 
a number of collectors leading ultimately to a single main, the 
same general principles may be applied. This, however, is a case 
not likely to arise. 

In the case of an interconnected network with inflow at one point 
and discharge at another, the general method of Problem 1 will 
still apply. It will be found that the number of junctions plus the 
number of possible paths of flow will equal the number of elements 
in the system, and thus a complete set of simultaneous equations 



58 HYDKAULICS OF PIPE LINES 

may be found which will serve to determine the velocity in each 
element of the system. In the case of Problems 2 and 3, the flow 
must be assumed or known in two out of each of the three pipes 
forming each junction. The flow in the third pipe will then become 
known from equations of the form 

This will then serve to determine the entire system of flow. 
Then by the arbitrary fixing of velocity, size or hydraulic gradient, 
in the same general manner as in Problems 2 and 3 above, the 
remaining features of the system may be determined. 



CHAPTER II 

THE PROBLEM OF THE SURGE CHAMBER 

22. General Statement of Problem and Derivation 
of Equations 

From (m) in Sec. 11, page 31, it appears that an unbalanced 
pressure head p/w acting over the cross sectional area of a pipe at 
one end will act as an accelerating force on the water in the pipe 
and will produce an acceleration measured by the equation : 

Ldv = p m 

g dt w K ' 

Now suppose that we have given the arrangement of Fig. 24 
comprising the following items : 

1. A supply reservoir R in which the water may be supposed to 

[remain at a constant level. 

2. A main conduit AB. 

3. A surge chamber or stand-pipe BG. 

4. A penstock line BL terminating in a control nozzle. 
Assume steady flow conditions with velocity in main conduit 

AB=v lf and level of water in surge chamber at C. This means that 
of the total head BG at B, the velocity head plus the friction head 
has absorbed an amount measured by GC, leaving the balance as 
pressure head measured by BG. Now the existence of steady 
conditions implies a balanced system of forces on the water in AB, 
composed as follows : 

1. Gravity with component acting A to B. 

2. Pressure at A distributed over cross section of pipe and 

acting A to B. 

3. Pressure at B distributed over cross section of pipe and 

acting B to A. 

4. Friction al resistance along pipe and acting B to A. 

Next suppose that with no change in the velocity v x or in the 
conditions generally in AB, there should develop suddenly a change 
in BG as a result of which the water level should drop to D. It is 
obvious that the force equilibrium of the water in AB would be 
correspondingly disturbed. The pressure head at B is now no 
longer BG but BD, an amount less by the head CD. The result 
will therefore be an unbalanced pressure head acting on the water 

59 



60 



HYDKAULICS OF PIPE LINES 



in AB in the direction A to B, and measured by this distance CD. 
This accelerating head will immediately operate to produce an 
acceleration in the velocity of flow along AB measured as indicated 
in (1). 

Similarly if with steady conditions and a velocity v x in AB, the 
level of the water in BG should suddenly rise to D lt there would be 
a corresponding disturbance in force equilibrium and an unbalanced 
pressure head measured by CD X would act at B over the cross 
section of the line in the direction B to A, producing a retardation 
in the velocity of flow along AB measured likewise as in (1). 

From this analysis the truth of the following general statement 
becomes self-evident. 

At any time during the period of velocity change let the velocity 




Fig. 24. — Problem of the Surge Chamber. 



along AB—v. Corresponding to this condition the velocity and 
friction heads combined will have a certain value, say GC. Then if 
the pressure head=^6 7 (that is, if the water level is at C) the 
conditions for force equilibrium will momentarily obtain, and there 
is no accelerating or retarding head in operation on the water in 
AB. If, on the contrary, the pressure head differs from BC (that is, 
if the water level is not at C) the conditions for force equilibrium 
will not obtain, and there will be in operation an accelerating or a 
retarding head measured by the distance of the actual water level 
below or above the level C. 

We may express this somewhat more briefly by saying that with 
any velocity of flow in AB, if the water level in BG is where it 
belongs for steady conditions, then there is no accelerating or 
retarding head in operation on AB, but if the water level is below 



THE PROBLEM OF THE SURGE CHAMBER 61 

or above the location for steady flow then there is in operation a 
corresponding accelerating or retarding head, measured by such 
difference in level. Or still otherwise ; the accelerating or retarding 
head acting on the water in AB is measured by the difference 
between the level of the water as it actually is and where it belongs 
for steady conditions with the given instantaneous value of the 
velocity. 

Now suppose a power house at L, the lower end of the penstock 
line, with fluctuating demand for power according to the accidents 
of the daily load curve. Suppose for the moment steady conditions 
with a velocity v x in AB and water level at C. Let there arise a 
sudden demand for excess power such as would require a rate of 
volume flow greater than that which the main conduit is furnish- 
ing. The nozzles at the power house, under governor control, will 
open up accordingly, the response from the adjacent penstock line 
BL will be prompt and we shall, after a few seconds time, have a 
flow through BL carrying the increased volume of water. The 
actual velocity in AB will, however, still remain substantially 
unchanged and the surge chamber BG must therefore supply the 
difference. As a result the level of water in the surge chamber will 
fall below the level for steady conditions, and as it falls will develop, 
as we have just seen, an accelerating head which will start in to raise 
the velocity of flow in A B from the initial up toward a higher final 
value. 

In a similar manner, it is clear that if load is suddenly rejected 
at the power house, corresponding to a decrease in the rate of 
volume flow required, then the surge chamber will receive, for the 
time being, the excess flow coming along the line AB, and as a 
result the actual level in BG will rise above the level for steady flow 
and a retarding head will be developed ; and as a result of which the 
velocity of flow in AB will be continuously and gradually reduced 
from the initial toward a lower final value. 

It thus appears that any sudden change in power demand will 
react on the surge chamber in such manner as to disturb the level of 
the water from its location for steady flow conditions, and thus to 
develop an accelerating or a retarding head. Following these 
conditions in some further detail it is seen that as the velocity v 
gradually rises, the velocity and friction head combination will 
increase and the level corresponding to steady conditions will begin 
to fall. We shall have as time goes on, therefore, a dropping actual 
level of the water and a dropping level for steady flow for the 
momentary value of the velocity v, the difference between these two 
levels measuring the accelerating head in operation. Thus in Fig. 25 
let OX denote a time axis, A the level of the water at the start of 
the change with velocity v x and D the level for the final steady flow 
velocity v 2 . Then the time history of the drop in the actual level 
might be some curve such as AEFD while that for the corresponding 
level for steady conditions with the momentary value of the 



62 



HYDKAULICS OF PIPE LINES 



velocity might be some curve such as ABCD. The differences 
between these, measured by the intercepts BE, CF, etc., give a 
time history of the accelerating head, as a result of the operation of 
which the velocity of flow will be finally raised from v x to v 2 . 

Similarly for rejected load ; the rise in the water level plotted 
on time might give some curve such as AEFD (Fig. 26), while the 




G II 

Fig. 25. — Acceleration Head. 



rise in the level for steady conditions with the momentary value of 
the velocity might give some curve such as ABCD. The intercepts 
BE, GF, etc., furnish then a measure of the retarding head as a 
result of which the velocity will be finally reduced from the higher 
to the lower final steady flow value. 




u H 

Fig. 26. — Retardation Head. 



The general problem of the surge chamber operating in connec- 
tion with a hydraulic power unit under governor control involves 
therefore the following conditions : 

1 . In the initial stage the power unit is assumed to be developing 
the power W x under steady flow conditions in the penstock and 
pipe line. While in this condition there is assumed to arise, let us 
say, a sudden demand for more power. 

2. In the final stage and after this demand has been satisfied, 
the power unit is again assumed to be developing the increased 
power W 2 under steady flow conditions in the penstock and pipe 
line. 

3. During the transition period, from (1) to (2) the governor is 



THE PROBLEM OF THE SURGE CHAMBER 63 

assumed to operate continuously to deliver to the wheel from the 
penstock, the flow of water required to develop the power W 2 * 
The rate of volume flow to the power unit will, during this period, 
not in general be the same as the rate of volume flow along the pipe 
line, and the surge chamber must therefore either supply or absorb 
the difference. This will occasion a change of water level in the 
surge chamber with a resulting accelerating or retarding head as we 
have seen. The change of level implies, however, a change in the 
pressure head at the surge chamber and hence in the net head 
ultimately available at the power house. But the rate of volume 
flow required to develop the power W 2 will depend on the net head 
available at the power house, and will approximately vary inversely 
as such head. Hence with demanded load, as the head continuously 
falls, the quantity of water required will correspondingly increase, 
while with rejected load, as the head continuously rises, the quantity 
of water required will correspondingly decrease. 

In the detailed discussion of this problem, consideration must be 
given to a number of different velocities, some actual and some 
virtual. These may be characterized and denoted as follows : 

v ± This is the actual conduit velocity under initial steady flow 
conditions. It is the actual velocity with the initial head 
and the original load W v 

u ± This is a virtual velocity and is really the measure of a rate 
of volume flow. It is the velocity along the main conduit 
which would supply the quantity of water which, under the 
initial head, would serve to develop the final power W 2 . 

v 2 This is the actual conduit velocity under final steady flow 
conditions. It is the actual velocity with the final head and 
final load W 2 . 
u This is a virtual velocity and is really the measure of a rate 
of volume flow. It is the velocity along the conduit which 
would supply the quantity of water actually required at the 
wheels to develop the power W 2 under the head which 
prevails at any instant during the transition period. Under 
final steady flow conditions u becomes the actual velocity 
v 2 . 
u m This is the maximum value of u. 

w This is the actual velocity along the penstock fine AB (Fig, 24), 
corresponding to the delivery of the quantity of water 
required by the wheels. 

Let A ± = cross section area of main conduit (f2). 
A 2 across section area of penstock (f2). 
#!=:nead GB, Fig. 24 (f ). 
# 2 =head BM , Fig. 24 (f ). 

* In the actual case this condition cannot be fully realized. The divergence, 
however, is not of significant importance in the treatment of the problem. 



64 HYDKAULICS OF PIPE LINES 

#=area of surge chamber at water level (f2). 
iy=length of conduit (f ). 

y= movement of water level from initial position (f). 
z= height of actual water level above B (f ). 
Let b x denote for the main conduit line the factor LjG 2 r (see 
Sec. 2) and b 2 the similar factor for the penstock line. We 
have then, 

b^^-^b^w^ =loss °f head due to friction under initial steady 

flow conditions. 
b 1 v 2 2 -\-b 2 w 2 2 —loss of head due to friction under final steady 
flow conditions. 
H—b 1 v 1 2 —b 2 w 1 2 == net head under initial steady flow conditions. 
H—b 1 v 2 2 —b 2 w 2 2 ==iiet head under final steady flow conditions. 

Referring now to Fig. 24, assume that at any instant during the 
transition period the actual level of the water in the chamber is at 
F while the level for steady flow with the existing velocity v is at E. 
Then the pressure head at B at this instant of time will be measured 
by z =BF and the total head will be BF-\- velocity head v 2 /2g. Hence 
we have z-\-v 2 /2g=&ctudil head at base of surge chamber under 
conditions prevailing during transition period. 

Also H 2 — 6 2 w; 2 = actual net head at power house available from 
H 2 . Hence z+v 2 /2g-t-H 2 — 6 2 w 2 =total actual head at power house 
under conditions prevailing during the transition period, with 
actual water level at F and surge or total movement of water level 
from C to F. 

Again, assuming substantially uniform efficiency for the hydraulic 
unit over the range from W x to W 2 , it results that the power 
developed will vary directly as the product of head by quantity per 
unit time or rate of volume flow. Hence we shall have the following 
expressions : 

v x (H — b^ 2 — b 2 w x 2 ) represents load W x under initial head. 
u ± (H — b^ 2 — b 2 w x 2 ) represents load W 2 under initial head. 
v 2 (H—b 1 V2 2 —b 2 W2 2 ) represents load W 2 under final head. 



u(z-\-y — \-H 2 — b 2 w 2 ) represents load W 2 under head prevailing 
^ during transition period. 

We may then derive the following relations between these various 
velocities : ,. a 

-A\ (2) 

Mfc (3) 

_ « 1 (ff-6 1 V-5 2 w 1 2 ) ... 

V *~ H-brt-b&S l ' 

u= MB-b 1 v 1 i -b t w 1 i ) {5) 



THE PROBLEM OF THE SURGE CHAMBER 65 

Regarding these formulse it may be noted that (4) is a cubic 
equation in v 2 . This, however, is readily solved by trial and error. 
Also in (5), which is intended to give u for known values of z and v, 
the denominator contains w, which is also unknown. We may there- 
fore proceed by assuming a value of w and then find u from equation 
(5) and check the result from (2). A trial and error procedure will 
thus readily serve to determine u for any stated condition of v and z. 
Or otherwise, by substituting for w in (5) from (2) the former may be 
reduced to a cubic equation in u and then solved by trial and error 
as with (4). 

We may now proceed to develop the equations for the flow of the 
water during the transition period. 

First for any velocity v in the main conduit we have : 

v 2 

_= velocity head. 

Lv 2 

— —=friction head (Sec. 2). 

C 2 r 

As above let E denote the level for steady conditions with the 

transition velocity v, and F the actual level. Then 

z=BF. 

CE- V \ Lv2 
GE -2g + C^r 

EF— accelerating head acting at the given instant. 
But EF=GB-(GE+BF). 
Whence from (1) we have 



L&v /v 2 Lv 2 \ 

~g dr Hl ~\2g^C 2 'r)~ Z ' 



Also (u— v)=deficit of velocity in AB. That is, the actual 
conduit velocity is v, while the velocity corresponding to the 
water required is u. 

Hence A (u—v)= deficit in rate of volume flow which must be 

made up by flow from the surge chamber. But 

dz 
—F—=r&te of flow from surge chamber. 



Then- Fj=A(u-v). 
Then we have finally 



ih 



- g ir^~ {cv2+z) (6) 



H.F.L. — F 



66 HYDRAULICS OF PIPE LINES 

For certain modes of study of this problem, equations (6) and (7) 
may be conveniently put into the form 

f,J=*-^-V) (8) 

zi=<-> • <*> 

These equations are readily seen to be the equivalent of (6) and 
(7). As written (8) and (9) apply to the case of demanded load 
and a falling level. The changes necessary for the case of rejected 
load and a rising level are made by changing the sign of dvjdt, by 
interchanging the signs of v 2 and v-f in (8) and of u and v in (9). 

Also from Fig. 24 the following relations are readily seen : 

z-^y—R^—cv^ (demanded load)) /^O) 

z—y=H 1 —cv 1 2 (rejected load) f ' 

Equations (6), (7) or (8), (9) with (2) and (5) serve to define 
mathematically the conditions of the movement of the water during 
the transition period. 

For demanded load with the actual level below the level for steady 
conditions, as at F, Fig. 24, (cv 2 -\-z) will be less than H 1 and dvjdt 
will be positive in sign representing an acceleration of the velocity, 
as is required to carry the value from v x to the higher v 2 . On the 
other hand, with rejected load and the actual level above the level 
for steady conditions, as at F 1? we shall have OE-\-BF 1 greater 
than H x and dvjdt will be negative in sign, representing a retarda- 
tion of velocity, as is required to carry the value from v x to the 
lower v 2 . 

Likewise in (7) with demanded load u > v and dzjdt is negative, 
implying a falling level, while with rejected load u < v and dzjdt 
is positive, implying a rising level. 

In the preceding discussion of the operation of a surge chamber 
and in the curves of Figs. 25 and 26 we have for simplicity assumed 
that the various quantities involved (water level, velocity and 
accelerating head) all move from the first or initial values to the 
final values continuously in one direction only, and hence without 
oscillation about such final values. Such movement is usually 
styled " dead beat," and is only sensibly realized with a relatively 
large size of chamber. In the usual case all three quantities, as 
above, approach and ultimately reach the final values or condition 
as the result of an oscillatory variation, of rapidly diminishing 
amplitude, about the ultimate values. Thus in the case of de- 
manded load (see Fig. 27) the water level will rapidly drop (AB) 
and the level for steady conditions with the momentary value of 
the velocity less rapidly, (^41), thus indicating an accelerating head 
in operation during this period. In the general case the water 
level will reach the steady motion level for final velocity v 2 (at B) 
before v has reached this value. In other words during the drop 



THE PROBLEM OF THE SURGE CHAMBER 



67 



AB, v is continuously less than u and dzjdt is negative, implying 
a falling level. At B this condition still continues and the level 
must drop still further {BG) until finally v=u m (level at C). At 
this point v has finally caught up with u (H) and dzjdt becomes 
This marks the lowest level reached by the water, a level also 
below that for steady flow (3) with the velocity u m . There is 
therefore still operative an accelerating head 30, which will carry 
the value of v beyond u m and still further beyond v 2 . This excess 



Movement of Water Level 




Fig. 27. — Time History of Movement of Water Level, Velocity 
and Acceleration Head 



of v over u will, however, bring more water than is required by the 
penstock and the result will be a rising level (CD). With the rising 
value of v will also come a further falling level for steady con- 
ditions, (3, 4), and the rising actual level will thus soon meet and 
pass that for steady conditions (at 4), thus changing the accelera- 
tion head from plus to minus. 

In the meantime under the influence of the accelerating head 
during the period ABC4: the velocity has been constantly rising, 
reaching the value v 2 at G, the value u m at H, and the maximum 



68 HYDRAULICS OF PIPE LINES 

value v m at /, where the acceleration changes sign. The sign of 
dz/dt, however, still remains the same, and the water level con- 
tinues to rise (4D1£), until finally, with water level at E the velocity 
again becomes equal to u at a value less than v 2 (J). In the mean- 
time the velocity decreases (IJ), while the level for steady motion 
rises 4, 5, 6. In this manner these various quantities see-saw 
back and forth, no two reaching the final steady motion values 
together until by a series of oscillations of rapidly diminishing 
amplitude, they all finally reach sensibly their ultimate values for 
steady motion with velocity v 2 . These various oscillations back 
and forth may be traced readily by the curves of Fig. 27, studied 
in connection with the above brief analysis of the first stages of 
the movement. It results that in the general case the various 
quantities pass through a periodic or oscillatory movement, strongly 
dampened, or with a rapidly diminishing amplitude, until after a 
number of swings, greater or less according to circumstances, the 
accelerating head becomes sensibly zero and the water level and 
velocity have sensibly reached their ultimate conditions. 

The extent of the first swing beyond final values (as at (7, Fig. 27) 
will depend on the size of the chamber relative to the other 
characteristics of the case. The smaller the chamber the greater 
will be the amplitude. With a chamber of sufficient size the swing 
beyond final values becomes negligible, and the movement is then 
sensibly dead beat. 



23. Treatment of Surge Chamber Equations 

Differential equations of the form presented in (6), (7) or (8), (9), 
and representing a dampened oscillation in which the retarding 
force varies as the square of the velocity, do not seem to permit of 
direct solution by any known mathematical means. Under these 
conditions the following courses are open : 

1. The treatment of restricted or special cases which may admit 
of mathematical solution. This represents a partial solution 
of the equations. 
' 2. The treatment by approximate methods involving some 
departure from the exact relations implied by the equations, 
and thus making possible a solution by direct mathematical 
means. 

3. The treatment of the equations as they stand or in simplified 
form by methods of approximate numerical integration. 
This method involves no departure in principle from the 
relations involved in the equations, and the only errors 
are those of numerical detail, and these may be made as 
small as desired. The direct treatment of the equations 
in this manner involves a trial and error process, and is 
necessarily tedious in numerical detail. 



THE PROBLEM OF THE SURGE CHAMBER 69 

4. The treatment of the equations as they stand or in simplified 

form by a reversed or indirect process of numerical inte- 
gration which avoids the trial and error feature of (3). 

5. The development, through an extension of the law of kine- 

matic similitude, of a series of relation coefficients serving 
to connect the actual case with a model of workable size. 
Observations are then made on the model, and these are 
transformed, through the proper coefficients, into corre- 
sponding values for the actual case. 

Space will permit of only a brief survey of these various methods 
of treatment. 

(a) Solution of Special Cases. 

Complete shut down with cylindrical surge chamber. Prof. I. P. 
Church* gives for the case of a complete and sudden shut down 
the following formulse : 

ev * =el)l *-y + * t (l-e- F ») (11) 



y m =cv 1 *+\ > (l-e- Fv ™) (12) 



Where y= movement of water surface measured from initial 
level (f). 
y m = maximum value of y or maximum surge (f ). 
v— velocity corresponding to any assigned value of 

y (fs). 

v 1 =initial velocity (fs). 
c=as in Sec. 22. 
e=Naperian base. 

P=— rF with the notation of Sec. 22. 
AL 

The value of y m is in a form which can only be solved by trial 
and error, but the successive approximations come very quickly 
and a sufficiently accurate value may soon be found. The author 
gives an illustrative case in which v 1 =7-5(fs), cv 1 2 =ll(f), c— -1955, 
F/A=2'25 } L= 140(f). Whence P= -2024.0 as a first approxima- 
tion y m was taken =20. Then finding the value of e— Py m we have 
1/57, and finding thence the value of the right-hand side of (12) 
we have 15-85 as the second approximation. Similarly assuming 
2/ m =16 on the right-hand side we find y m =l5-l, and again putting 
y m =l5-l on the right-hand side we find ^=15-74, which is suffi- 
ciently close check. By actual experiment in this case a value 
y =16 was observed. 

Complete shut down with overflow. For this special case Prof. 
I. P. Church gives formulae as follows.*)" (Notation transformed to 

* " Cornell Civil Engineer," Dec, 1911, p. 114. 
•j- " Cornell Civil Engineer," Jan., 1914, p. 156. 



70 



HYDKAULICS OF PIPE LINES 



correspond with that used in the present work. See Sec. 22 and 
Figs. 28, 29). 

Let x= distance moved by water in main conduit after beginning 
of overflow (f). 
J^=volume of overflow (f3). 
#!= velocity in main conduit at instant of shut down (fs). 




1 



Fig. 28. — Complete Shut-Down with Overflow. 

#2= velocity in main conduit at beginning of overflow (fs). 

Found by (11). 
v= velocity in general (fs). 
t=time from beginning of overflow (s). 

=N /^ (for Kg. 28). 

17. 17 

(for Fig. 29). 




' u 




Fig. 29. — Complete Shut-Down without Overflow. 
Then for Fig. 28 : 



x== ic log < 

' = 2-|s l0g < 



and for Fig. 29 : 



k— a— cv 2 2 ~\ 
k—a—cv 2 

_{v-S)(v 2 +S)] 

2gc gt [a-lc+cv* ] 



: =^[ tan " 1 Ir tan " 1 i] 



(13) 
(14) 

(15) 

(16) 



THE PROBLEM OF THE SURGE CHAMBER 71 

If T=time to bring v to zero (end of first up surge) then we have 
T=A-tan-^ 2 (17) 

These equations give the distance x, the overflow Ax and the 
time t, all corresponding to any assigned- value of v lying between 
v 2 and S for Fig. 28, or between v 2 and for Fig. 29. 

In the case of Fig. 28, S is the final velocity, and from (14) the 
time required to reach this velocity will be oc. In the usual case, 
however, the approach to the final velocity is rapid, and the final 
condition is sensibly reached in a relatively short period of time. 



Scale of v 

3 4 5 



6 



8 9 10 




j? 1Qt 30. — Diagram of Movement of Water Level and Velocity 

(Opening). 

Full opening from closure. Church's formula.* If in equations. 
(8) and (9) v x is taken as zero and neglecting governor action we 
put u=v 2 and then combine the two equations so as to eliminate 
dt, we find the equation : 

AL 

ydy-cv 2 dy=y-(v 2 -v)dv. 

Integrating between limits of and y m for y, and and v 2 for 
v we have 






Fg 

"Trans. Am. Soc. C.E., 1915," Vol. LXXIX, p. 272. 



(18) 



72 



HYDRAULICS OF PIPE LINES 



The integration indicated in the second term cannot be carried 
out unless the relation between v and y is known. In some cases 
the assumption that the relation is that of the quadrant of an 
ellipse, as indicated in Fig. 30, has been found to give results closely 
agreeing with observation. 

On this assumption, the half axes being taken as v 2 and y m equa- 
tion (18) reduces to ^^ 



y* m -0'miev 2 *y n =j=-v. 



Whence 

2/ m =-096ct; 2 2 + 

or approximately 

K 



Fg 



/AL 

«W Fg 



-f-0092c 2 v 2 



(19) 
(20) 



(21) 



'AL 

10 ' " a V Fg 

Investigation through numerical integration shows that where 
values of AL/Fg are small, that is, where the surge chamber is large 



8t 
70 
60 

50 

z>40 

4i 
»■«> 
o 
#20 

JO 


_ . 




-^ 


^•^ 
























<^ 


, 




















N: 


\ v * 


















X] 


\\ 










iC^r- 


— - 








\ 


% 
















^ 




Srf 






































u 


i 


' 2 


<j 


* 4 


i 


' 6 


5 


b 


i 


10 



Scale of v 

Fig. 31. — Diagram of Movement of Water Level 
and Velocity (Closure). 



and the movement of the water surface nearly dead beat, the above 
assumption regarding the form of the relation between v and y 
is not closely realized, and the application of the formula will 
involve a considerable error in the resultant value of y. Its use 
cannot, therefore, be recommended in such cases. 

Full closure. — If the assumption made by Church regarding the 
relation between friction head and velocity be extended to the case 
of closure (see Fig. 31) the equation corresponding to (21) takes 
the form 

, I - ' -arr h( f ) C 



Fg 



THE PKOBLEM OF THE SURGE CHAMBER 73 

Numerical investigation here, likewise, shows that the formula 
is not applicable in cases where ALjFg is small. 

In a particular case where ALjFg had the value 43-5 with h 
for full velocity 32 and a range of velocity from to 10, the applica- 
tion of numerical integration gave the following results : 
For opening, v— to v =10, y m =69-Q. 
For closure, v=10tov=0, y TO =78-6. 
The application of the above approximate formulae gives corre- 
sponding results as follows : 
For opening, y m =69-2. 
For closure, y m =77*5. 
Again with the same maximum value of h and the range of v the 
same as above but with AL/Fg=4:'35, the application of numerical 
integration gives results as follows : 

For opening, v=0 to v=10, y m =32-0. 
For closure, v=10 to v=0, y m =38-7. 
The application of the above approximate formulae gives corre- 
sponding results as follows : 
For opening, y OT =24-06. 
For closure, y m = 34-07. 
In the first case the results by formula give a close approximation 
to the correct values. In the second case the error is more con- 
siderable. Examination in the latter case shows that with the 
values as taken, the relation between v and y bears no very close 
relation to an ellipse and hence the results derived on such an 
assumption are naturally in error. In Figs. 30, 31 the full lines 
show the actual form of the curve between v and y for the two cases 
as noted, while the dotted lines show the corresponding ellipses. 
The degree of departure is thus plainly apparent. 

For the case with full closure, Johnson* proposes the formulae : 

"Jtj+' v < 23) 

-iV^W *> 

(£=time to reach y). 

Tests of all such formulae, through approximate integration, 
show that they are applicable each to a relatively narrow field of 
use only and that application outside of such range may lead to 
considerable error. 

(b) Approximate General Solutions. — The following approximate 
equation is given by Johnson. f 

Vm*^W-*J*+*W M -*i*)* (25) 

* "Trans. Am. Soc. C.E., 1915," Vol. LXXIX, p. 265. 
t "Trans. Am. Soc. Mec. Eng., 1908," p. 455. 



Vi 



74 



HYDRAULICS OF PIPE LINES 



In this form the equation applies to the case of demanded load. 
For the case of rejected load the form is the same but with the 
interchange of v 2 and v v This implies equal values of y m for the 
two cases. In this equation the value of v 2 is to be taken somewhat 
greater than v 2 for the case of demanded load and less than v 2 in the 
case of rejected load. The author of the formula states that for the 
case of demanded load the value may be taken from the equation : 

<=£, w 

Where H is the total head and y is a first value of y m derived 
from (24) by taking v 2 —v 2 . For rejected load the inverse ratio 
(#— -?/)/# may be correspondingly inferred, although the author 
does not specifically refer to this case. 

As a further refinement or check on these equations, with special 
reference to demanded load, Larner* proposes the following : 



y, 



(V-"i) 2 +c 2 « -*i 2 ) 



Fg 



•%)• 



(27) 
(28) 



AL 

Fc 

2000 

4000 

6000 

8000 

10000 

12000 

14000 



TABLE XXIII 

AL 



k 

•50 
•62 
•67 
•70 
•73 
•75 
•77 



Fc 
16000 
18000 
20000 
22000 
24000 
26000 
28000 



k 
•79 

•80 
•81 

•82 
•82 
•83 

•83 



To use these equations proceed as follows : 

1. Use equation (25) and find a trial value of y m ; or otherwise 

assume a value according to judgment. 

2. Substitute this value in (27) and find v 2 . 

3. From the specified load change find u x from (3). 

4. From the known characteristics of the case find the value of 

ALjFc and using this in Table XXIII find the corresponding 
value of k. 

5. Substitute the values of v 2 \ u x and k in (28) and find u m . 

6. Substitute this in (5) (see also (2)) and find z. In substituting 

in (5) v 2 /2g which is unknown may safely be approximated. 
This value of z with (10) will then give y m . This is then 
compared with the assumed value. 

7. Correct first assumption and proceed as before by trial and error. 
These equations and table rest on an empirical basis derived 

from the examination of the results for fifteen selected cases 
* "Journal Am. Soc. Mech. Eng., Jan., 1909," p. 113. 



THE PROBLEM OF THE SURGE CHAMBER 75 

through the method of arithmetical integration, as explained by the 
author in the reference cited. 

(c) Problems as Simplified by Disregarding Governor Action and 
by Assuming Friction Head to Vary with First Power o! Velocity.— If 

the volume of water delivered to the wheel during the transition 
period be assumed as constant at the rate A ± v 2 equations (2)-(5) 
disappear from the problem and we have left simply (6) and (7), in 
the latter of which u becomes v 2 . This is equivalent to a disregard 
of governor action during the transition period and to the assump- 
tion of a delivery of a constant volume flow of water instead of the 
development of constant power. 

While this still leaves (6) and (7) beyond the reach of direct 
mathematical solution it nevertheless aids materially in treatment 
of the problem in various approximate or special or indirect ways. 
While the amount of surge as found for this simplified case is 
smaller than the true value, the magnitude of the error is often not 
serious. Especially is this the case where the friction head is a 
small fraction of the total head at the power house. On the other 
hand, where the friction head forms some considerable part of the 
total head the error resulting from the use of the simplified form 
might become significant. 

In particular this simplified form of the equations is well adapted 
to treatment by an approximate method which has attracted the 
attention of a number of writers on this subject.* This method is 
based on the assumption that the f rictional resistance in a pipe varies 
with the first power of the velocity instead of with the square. 
As a result of this assumption, equations (6), (7) becomes simplified 
in such manner as to admit of direct mathematical treatment. 

Following are some of the more useful results. 

Rejected load, partial or complete closure. — Assume in general the 
notation of Sec. 22. Also the following : 

A v = (vj — v 2 ) = velocity change . 

Ah=cAv= difference in friction heads for v x and v 2 on 
assumption that h varies as v. 



—M. 




-va- 


-a 2 


B Ai- 


-ac ) 


m 




4 U m 

tan0= 




a 




1 m 





* Prasil, " Schweizer Bauzeitung," Band LII, No, 21-25. Warren, 
Trans. Am. Soc. C.E., 1915," Vol. LXXIX. 



76 HYDKAULICS OF PIPE LINES 

Then y— Ah=e~ at (B sin mt— Ah cos mt) (29) 

y m —Ah=e- at i(B sin0— Ah cos 6) (30) 

Equation (29) gives the value y for any value of t. 
Equation (30) gives the value of y m in terms of an angle and the 
corresponding time t l9 determined as above. 

Numerical Example : 

£=9000 (f ). 

1^=5000 (f2). 

A=75 (f 2). 

« 1= 6-5 (fs). 

fc 1= =9-5 (f). 

v 2 =0 (complete shut down). 

We then find by substitution of numerical values : 
Av=fr5. 
Ah=&5. 
c=9-5/6-5=l-4615. 
a=-002611. 
m = ,00684. 
5=10-63. 
tan 0= -2-620. 

0=llO°-53'=llO°-9=l-9355. 
«!=283 sec. 
sin 0=-9343. 
cos 0=— • 3565. 

a^=-7388. 
e-«<x=.4775. 
B sin 0=9-929. 
Ah cos 0=3-a87. 
diff. = 13-316. 
e- a< xXdiff.=6-36. 
add 9-5. 
2/ m =15-86. 

While the above equations in the form given apply specifically to 
the case of rejected load and reducing velocity, the numerical 
values, on the present hypothesis regarding the relation of friction 
head to velocity, are the same for change in either direction 
between the same limits of velocity. Any problem in demanded 
load may therefore, for numerical solution, be converted into the 
corresponding problem in rejected load (change between the same 
velocity limits) and the solution found by the equations as above. 

The error introduced by the present assumption regarding the 
relation of friction head to velocity is such as to give by the result- 
ing equations a result somewhat too small for rejected load and 
somewhat too large for demanded load. In the case cited earlier, in 
comparing the results given by equations (21), (22) with those 
given by numerical integration, the value given by the presnet 



THE PROBLEM OF THE SURGE CHAMBER 77 

equations is 74 for either rejected or demanded load, lying nearly 
midway between the values 78-6 and 69-6 found by numerical 
integration. 

(d) Treatment of Surge Chamber Equations by Methods of 
Approximate Integration. Surge Chamber not Limited to Plain 
Cylindrical Form. — In the methods of treatment thus far discussed, 
the surge chamber is assumed to be of the plain cylindrical form 
with uniform cross section. The assumption of a form of surge 
chamber with varying cross section, as for example, a tapering or 
conical form, would introduce further complications beyond the 
reach of methods of this character. 

In the method of the present section, however, there are no 
such limitations, and the surge chamber may be of any form 
desired. 

Writing again equations (6) and (7) we have 

fir*.H-+-) .-w 

F dz . 
-A1= (W - V) (7) 

If the surge chamber is of the plain cylindrical form, the ratio 
FjA is constant. If the chamber is of varying cross section, then 
FjA will vary according to the known values of F at the water 
level. 

Suppose now that at any given instant of time t, all values of the 
variables, dvjdt, v, z, u and dzjdt are known. Next take a small 
interval of time At and for the time t-\-At assume a value of the 
acceleration dvjdt. We thus have two values of the acceleration, 
one at the beginning and one assumed at the end of the time interval 
At. If At is small we may take, without serious error, the mean of 
the two values multiplied by At as a measure of Av, the change in 
velocity for the same time interval. This will give a value for the 
velocity at the end of this time interval, or for time t-\-At. Next 
putting in (6) the assumed value of dvjdt and the derived value of v 
(both for time (t-\-At)) .we may solve and find the value of z for the 
same instant of time. 

Taking now the new values of z and v with an assumed value of 
w in (5) we find by trial and error the correct value of u for the 
derived values of z and v. This value of u substituted in (7) will 
give the corresponding value of dzjdt. We then have two values of 
dzjdt (for the beginning and end of the interval At), and using the 
mean with At we find the resulting value of Az, the change in z for 
the period At. Combining this with the initial value we find a 
value of z for the time t-\-At. We have thus found values of z for 
the end of the interval At in two different ways, one by way of (6) 
and one by way of (7), both starting from the assumed value of 
dvjdt at the time t-\- At. If this assumed value was correct, the two 
values z will agree. If incorrect they will disagree. The latter will, 



78 



HYDRAULICS OF PIPE LINES 



of course, be the result in the usual case. This operation must there- 
fore be considered as giving a first approximation. The extent and 
direction of the difference in the two values of z will serve to 
indicate the nature of the change to be made in dvjdt by way of a 
second approximation. In this manner by successive steps, a set of 
values for all the variables is found which will satisfy the equations 
(6) and (7) ; and such are then taken as the values for the instant of 
time t+At. The operation is again repeated for another inter- 
val At , and so along step by step as far as it may be desired to trace 
the history of the movement. 

As noted, the method requires a starting-point at which all 
values of the variables are known. This is the case at the beginning 
of the period of acceleration in the main conduit. At this instant, 
assuming the velocity in the penstocks to have acquired the value w x 
(corresponding to u x in the main conduit) and with the correspond- 
ing beginning of movement in the level of water in the surge 
chamber, we shall have dv/dt=*0, v=v 1 , z=H—bv 1 2 and dz/dt= 
— {% x —v^)A jF. These then are the initial values from which a start 
may be made in the manner described. 

If desired other and more accurate rules for numerical integration 
may be employed, as for example, the following : 
a=ft/12 (5y 3 +8y. 2 - yi ). 
For the area between the two ordinates y 2 and y 3 of Fig. 32. This 
assumes the arc of the curve to be a 
second degree parabola and is therefore 
more accurate than the use of an arith- 
metical mean, which assumes the arc of 
the curve to be a straight line and the 
contour to consist of broken straight 
lines. This rule can be used as soon as 
two sets of values of the various quanti- 
ties are known ; that is, as soon as the 
first point beyond the initial has been 
found in the manner above indicated. 
In carrying out this work it is highly 
advantageous to carry along a series of plotted values of the 
different variables. In this manner by noting the trend of the 
curve the new value of dvjdt to be assumed in each case may 
be chosen very near to the correct value, and a satisfactory set of 
results therefore determined with the minimum number of trial 
and error steps. 

The details of the work may be modified in various ways, as the 
interested reader will readily discover for himself. 

By varying the time interval At according to the conditions of 
the problem any desired degree of accuracy may be realized. 
According to the dimensions and conditions involved and the 
degree of accuracy desired, the value of At may be taken from 
5 see. or less to 100 sec. or more. 




Fig. 32. 

Rule for Approximate 
Integration. 



THE PROBLEM OF THE SURGE CHAMBER 79 

If instead of treating (6) and (7) in connection with (2)-(5), the 
simplified case is taken, assuming constant volume discharge of 
water to the wheels, then, as noted previously, u becomes constant 
=v 2 and the details of the process are greatly shortened. 

(e) Treatment of the Surge Chamber Problem through the 
Assumption of a Predetermined Program of Acceleration. — The 
method developed in the preceding section involves necessarily a 
trial and error process. This results from the fact that the dimen- 
sions and proportions of the chamber are assumed as fixed, while 
the quantities to be determined are the consequences developing 
from a stated change in the load conditions. 

If, on the other hand, the dimensions and proportions of the 
chamber are left to be determined, then we are free to fix according 
to choice any of the other variables entering into equations (6), (7), 
as, for example, the time history of the acceleration dvjdt. This is 
equivalent to stating the problem thus : Given a program for the 
water, required a surge chamber to produce it. The problem thus 
stated admits of solution by a straightforward process and without 
trial and error approaches. 

For the details and possibilities of this method, reference may be 
made to a paper by the present author,* in which a full discussion 
of the subject will be found. 

It may be here noted, however, that by this method there is, of 
course, no assurance that the form of chamber resulting from any 
arbitrarily assumed time history of the acceleration will be accept- 
able from a structural view-point. While it might produce the 
particular program of acceleration proposed, yet the form or 
dimensions might be undesirable. It results practically, however, 
that with a little experience the nature of the relation between the 
curve assumed for acceleration and the resulting form of the 
chamber comes to be readily appreciated, so that a form of curve 
suited to any generally proposed form of chamber may be assumed 
without difficulty. 

It may be also noted that by this method, the minimum dimen- 
sions of chamber which will render the movement of the water 
substantially " dead beat " are readily determined. 

(/) Treatment of the Surge Chamber Problem by Model Experi- 
ment through the Application of the Law of Kinematic Similitude. — 
Let L, d, D, A, F, v and H denote, as in Sec. 22, length of main 
conduit, diameter of main conduit, diameter of surge chamber, area 
of main conduit section, area of surge chamber section, velocity in 
main conduit and head in general. 

Now let us assume the existence of two different cases with 
different values of these various characteristics, but so related as to 
produce, for similar load changes, similar time histories of the 
acceleration head in the surge chamber (EF„ Fig. 24) and hence 

* "Trans. Am. Soc. Mech. Eng.," Vol. XXXIV, p. 319. 



80 



HYDRAULICS OF PIPE LINES 



similar histories of the acceleration of the velocity in the main 
conduit. 

The term similar, as here used, implies the transformation of one 
history into the other by a suitable change in the scales for accelera- 
tion head and for time. Assuming the time histories plotted as 
curves, this will imply the transformation of one curve into the 
other by a suitable change in the scale of the horizontal and vertical 
axes. Thus in Fig. 33 let the curve OA 2 B 2 represent the accelerating 
head for case No. 2 plotted on time. Then if the vertical ordinates 
or dimensions of OA 2 B 2 are multiplied by a factor, say -60, and the 




Fig. 33. — Similar Curves. 



horizontal dimensions by a factor, say -80, another similar curve will 
be produced as shown by 0A 1 B V Thus to every point on No. 2, 
as for example, P 2 , A 2 , Q 2 , etc., there will correspond a point on 
No. 1, as P lt A lf Q v etc. 

Writing now equations (6) and (7) we have 

^H.-icv^+z) (6) 



g dt 
Fdz 
Adt~ 



(u—v) 



(V) 



The left-hand term of (6), as we have seen in Sec. 11, is a measure 
of the acceleration head. Now assume this equation applied 
numerically first to case No. 1 and then to No. 2. At corre- 
sponding instants of time the left-hand members will be related 
by a fixed ratio. Hence the same must be true of the right-hand 
members, term by term. 

Without attempting here to develop any general discussion of 
the principles of kinematic similitude, it may be noted that in the 
case of any algebraic equation which is to be applied to two different 
physical systems involving similar phenomena, such application 
being made effective through the use of a scale or transformation 
ratio, then each term in the equation must be subject to transforma- 
tion from one system to the other through the use of the same scale 
ratio or factor. Or in other words the equation must be homo- 
geneous in the transforming factor. The underlying reason for this 
may be seen by referring again to (6). The left-hand member is a 
quantity of the order of a vertical height ; it is a linear dimension. 



THE PROBLEM OF THE SURGE CHAMBER 81 

Hence every mejnber of (6) must also be a linear dimension and 
whatever ratio may exist for any one member as between case 1 and 
case 2 must also hold for all other members. 

We are now ready to determine the various relation factors 
between the two cases in order that the assumed condition of 
similarity between the two acceleration curves may be fulfilled. 
Let the length ratio or L 2 /L 1 =p. 
Friction plus velocity head or c ratio =c 2 lc 1 =q. 
Velocity ratio for any two corresponding velocities —r. 
Time ratio for any two corresponding periods of the move- 
ment =s. 
Then we have immediately as follows : 

v 2 ratio =r 2 . 
cv 2 ratio =qr 2 . 
But cv 2 is a term in (6) and represents therefore a vertical dimen- 
sion. Hence the same ratio qr 2 must hold between all other terms 
of (6) and in general between all similar vertical dimensions in the 
two cases. Hence we have 

H ratio —qr 2 
z ratio —qr 2 
y ratio =qr 2 

and — 7- ratio —qr 2 . 
g dt 

But the L ratio =p and hence 

dv . . qr 2 
-T- ratio =2— 
dt p 

But the dvjdt ratio must equal the quotient of the velocity and 

time ratios or rjs. Hence we have 

r _qr 2 

s p 

p 

or s=— 

qr 

Again the dzjdt ratio must equal the quotient of the z and t 



ratios, or 



dz or 

■v- ratio =— or substituting the value of 6* 

dt s 

dz . . q 2 r s 
-T- ratio =- — 
dt p 



Hence from (7) we have 



H.P.L.— G 



F q 2 r* 

-r ratio X - — =v ratio =r. 

A p 

F • . p 

or -r ratio =-5-5 
A q 2 r 2 

or F ratio = -f- 9 X A ratio . 
q 2 r 2 



82 HYDKAULICS OF PIPE LINES 

But F ratio = (D ratio) 2 = (D.JDJ* 
and A ratio —(d ratio) 2 =(d 2 /d 1 ) 2 
so that we have finally : 

\/~V 

D ratio = *-£ X d ratio. 
qr 



Pw^«^XI 



Volume ratio =jP ratio X height ratio =- X (d ratio) 

Collecting for convenience we have a series of ratios as follows : 

(a) The ratio of all similar vertical dimensions, such as maximum 

movement of water level, movement for final steady 
conditions, movement for corresponding parts of the 
acceleration curve or in corresponding values of the time, 
will equal qr 2 . 

(b) The ratio of the time intervals for the whole or for corre- 

sponding parts of the transition phenomena will equal — , 

(c) The ratio of the diameters of the chamber at corresponding 

levels will be & (S) or 3^/5 

qr \dij qr V A x 

(d) The ratio of the volumes swept through by the water surface 

mi i V d 2 2 V A 2 
in corresponding times will be - -f- or - -j*. 

q a x q A x 

It results that if these various relations between the dimensions 

and characteristics of the two cases are fulfilled, then the two 

acceleration curves will be similar ; or otherwise, for similar velocity 

changes the acceleration curves will be similar and the movement of 

the water level in the surge chamber for one case will be in a known 

relation to that in the other case. 

The application will be made clear by an illustrative case. 

Let £ 2 =30,000 (f). 

d 2 =S (f). 

a 2 =50-27 (f). 

Upper velocity =8 (fs). 

Friction+velocity head at 8 (fs)=66-7 (f)> 

Value of c 2 =l-042. 

Proposed diameter of surge chamber =45 (f). 

Suppose now that it is desired to forecast the behaviour of this 

case by means of a model using for the conduit say 20 feet of pipe, 

1-inch internal diameter. 

Then length ratio =p= 3 0>CK)0/20= 1500. 

Diam. ratio =^ 2 /^=96/l =96. 

Suppose that it is found by experiment that an upper velocity 

of 5 (fs) will be most suitable for the model. Then velocity ratio 

r=8/5=l-6. Next let the coefficient c be determined for the pipe 

by experiment. Suppose for a velocity of 5 (fs) the friction -f 

velocity head is 2*5 (f ). 



THE PROBLEM OF THE SURGE CHAMBER 83 

Then for the model c 1 =2-5/25=-l. 

Then for the c ratio we have 2=l-042/-l = 10-42. 
Then we have as follows : 

From (b) the time ratio 5=89-98 

From (a) the vertical ratio =26-68 

From (c) the ratio D 2 /D 1 =223 

Whence D^^X 12)/223=2-42 (i). 

Suppose then that we fit up the pipe with a model surge chamber 
of diameter 2-42 (i) and observe the movement of the water under 
various conditions of load change. 

Thus, for complete shut down full flow let the total rise of water 
be 35 inches or a 5-inch surge beyond the position of final equili- 
brium. Then the corresponding movement in case 2 should be 
5x26-68=133-4 (i) or 11-1 (f). 

Again, if the time to maximum rise of water level is 6-5 (s) in 
the model, the time in case 2 should be 6-5x89-98=585 (s)=9-75 
minutes. 

Again, if it be desired to know the movement of the water level 
for an increase in load corresponding to a change in velocity in the 
main conduit of case 2 from 3 (fs) to 6 (fs) we observe the move- 
ment with the model for a change from 1-875 (fs) to 3-75 (fs), and 
apply the suitable ratios. 

24. Differential Surge Chamber 

The differential surge chamber consisting, in effect, of a small 
riser or stand-pipe connected to the line and to the penstock and 
standing within a larger chamber to which it is connected with 
ports, has been investigated and discussed exhaustively by John- 
son.* Limitations of space forbid a detailed discussion of this 
combination of elements, but the interested reader should refer 
to the original papers, as noted, for a full consideration of character- 
istics offered by this device. 

* "Trans. Am. Soc. Mech. Eng., 1908," p. 457. "Trans. Am. Soc. C.E.," 
Vol. LXXVIII. 



CHAPTER III 

WATER RAM OR SHOCK IN WATER CONDUITS 

Assume a straight inclined conduit, as in Fig. 34, leading from a 
reservoir at the upper end and controlled by a valve at the lower 
end permitting of opening or closure at any desired rate. 

Suppose the water flowing steadily with a given conduit velocity v. 
If now the valve B is suddenly opened or closed, wholly or partially, 
there will be initiated at B a disturbance in the pressure condition 
of the water of the same nature as an acoustic wave in air, and 
subject to the same general laws of propagation. 

As the first specific case we shall consider complete and instan- 
taneous closure. 

25. Water Ram with Instantaneous Complete 
Closure : General Physical Conditions 

In this case we may picture the physical conditions as represented 
by a moving elastic column of water suddenly arrested at the lower 
end. We shall further, for simplicity, first assume the pipe as rigid 
and disregard the influence of friction on the pressure head of the 
water. After developing the physical conditions presented by this 
relatively simple and ideal case, the modifications necessary to 
allow for elasticity of the pipe and for friction will receive con- 
sideration. 

By analogy we may picture a long spiral spring, moving endwise 
with a velocity v, and suddenly striking an immovable obstacle 
at the forward end. The result for the water column will be a 
compression beginning at B and propagating upward toward A, 
until finally the entire column will be brought to rest in a condition 
of compression from one end to the other. The period of time 
required for reaching this phase is, furthermore, just the time re- 
quired for the propagation of an elastic compressive wave the length 
of the line, or from B to A. Physically we may picture the lower 
end of the column coming to rest against the face of the valve 
and the remainder of the column continuing on until at each point, 
as it reaches the same degree of compression, it also comes to rest ; 
and thus the edge of the compressed section travels up the line with 
the velocity of propagation of an acoustic wave, until at the close 
of the period, the entire mass of the column is at rest under com- 

84 



WATER RAM OR SHOCK IN WATER CONDUITS 85 

prespon. In Fig. 35 let A B denote the length of the column at 
the instant of closure, and the arrow the direction of motion. Then 
A 1 B 1 will denote the compressed length of the column at rest, or 
the condition at the close of the period winch we have just con- 
sidered. Again, it will be clear that the kinetic energy which the 




Fig. 34. — Shock in Pipe Lines. 



i- A* 



Aim 



Bo 



4.J 



column of water possessed in virtue of its velocity v, has, at the end 
of this period, disappeared and become transformed into the 
potential energy of compression under 
the condition denoted by A 1 B V In the 
condition A X B^ therefore, while the column 
of water is at rest momentarily, it is not 
in equilibrium with its surroundings, since 
it is under the excess pressure resulting 
from the compression as noted. It will, 
in consequence, begin to return toward 
normal condition, and a wave of expansion 
will start in at the upper end A x and 
progress downward toward B v The com- 
pression, in other words, will gradually 
yield, the particles of the column moving 
upward and the wave of expansion pro- 
gressing downward until finally, in the 



u 



B, B 2 Bj R. 

Fig. 35. 

Successive States in 

Oscillating Water 

Column. 



phase A 2 B 2 , the original condition will be 
reached, but with the particles of water 
moving upward. Comparing the con- 
ditions AqBq and A%B 2 , we have the 
volume and condition as regards pressure 
the same ; also the energy is entirely kinetic in both cases, but 
reversed in direction in A 2 B 2 as compared with A B . 

Considering A 2 as virtually a free end, it is clear that the result 
will be the same as though the column as a whole should continue 
to move upward, thus relieving the pressure at B 2 , and starting 
in a wave of further dilatation at B 2 which will progress upward 
toward A 2 . The ultimate result will be a condition A 3 B 3 , in which 
the column will be momentarily at rest in a state of dilatation. 



86 HYDRAULICS OF PIPE LINES 

The kinetic energy in the state A 2 B 2 will thus be stored up against 
the gravity forces which serve normally to produce the condition 
of relative compression denoted at AqB or A 2 B 2 . It will be shown 
later that as compared with the normal pressure in A 2 B 2 , the drop 
in pressure in A Z B Z will equal the rise in A X B V 

Again, while this is a condition of rest, it is not one of equilibrium, 
and it will be followed by a return wave of relative condensation, 
beginning at A z and progressing downward toward B z . The result 
of this will be a return toward normal condition, and ultimately 
the column will be in the state AJB^ with the energy kinetic in 
form and the particles travelling downward, and with the original 
state as regards volume and pressure. 

Condition A i B i is thus seen to be the same as condition AqB . 
We have thus traced the phenomena through a complete cycle, 
and assuming a perfectly elastic liquid, the sequence of compression 



Btn 



m 




5 6 7 8 9 10 



11 12 



Fig. 36. — Successive States in Oscillating 
Water Column. 



and dilatation in volume with rise and drop of pressure above and 
below the normal, would continue indefinitely. Actually due to 
viscous resistance the oscillations will dampen out after a few 
cycles, the number and character depending on the circumstances 
of the case. 

As a further aid in picturing the condition of the water in the 
various phases of this wave of compression and dilatation, reference 
may be made to Fig. 36, showing intermediate phases, with the parts 
of the column under compression or dilatation indicated by the 
shading. 

If now we assume a pressure gauge located at the lower end of 
the line, it is clear that relative to normal pressure the phases from 
to 6 (Fig. 36), will show a rise in pressure and from 6 to 12 a drop 
in pressure. This is shown by the line OABCD (Fig. 37). This 
diagram, of course, assumes ideal conditions in the liquid and a 
gauge acting without lag or inertia. It thus appears that the 




WATER RAM OR SHOCK IN WATER CONDUITS 87 

pressure will suddenly rise to an amount OA above the normal 

and hold such value uniform during the phases to 6 (Fig. 36), at 

the end of which period it will 

suddenly drop to the point G 

(Fig. 37), below normal, which 

value it will hold during the 

phases 6 to 12 (Fig. 36), following 

which there will be a reversal to 

excess pressure and a repetition 

of the cycle. 

From a study of the diagram 
(Fig. 36), it will also be clear that 
at any point in the line not at Fig. 37. — Time History of Excess 
the lower end, as at G, the com- Pressure at Valve. Ideal Case. 
pressive phase will not begin 

until the wave has travelled from A up to G, and that it will 
continue only during the time required for the wave to travel from 
G to the upper end B and return to G. 
In general let 

L=length of line. 

#=distance to any point from lower end. 
#=velocity of propagation of acoustic wave. 
Then 

#/#=time interval after closure to beginning of com- 
pression. 
2 (L—x)/S= duration of period of compression. 

x/S=time interval from close of period of compression 
to end of cycle. 
Similar expressions will hold for the phases of dilatation. 
For a point near the lower end, or where x is small, the pressure 



) 

1 




• 
















• 


1 


2 I 


\ 4 
* 


5 














i 



Fig. 38. — Time History of Excess Pressure at Points 
in the Line. Ideal Case. 



diagram will therefore be similar to Fig. 38, while for a point near 
the upper end, or where x is large, it will be similar to 39(a), and 
for the upper end itself it will be similar to 39(6). 

These various diagrams may be represented as a system by the 
solid diagram suggested in Fig. 40. FGHJ represents a reference 
plane corresponding to normal pressure. A is a wedge lying on 



88 HYDRAULICS OF PIPE LINES 

the upper side of the plane and G a similar wedge lying on the under 
side of the plane. The triangles D, B and E are parts of the plane. 
The lower edge FJ corresponds to the lower end of the pipe, the 
upper edge GH to the upper end, and any intermediate cutting plane 
11, 22, etc., to a point distant x from the lower end. Then any such 



J"! 



Fig. 39. — Time History of Excess Pressure 
at Points in Line. Ideal Case. 



plane 11, 22 or 33 will cut from such a model a section, the out- 
line of which will give the diagram of pressure for the corresponding 
point in the pipe line. 

We have now to show that the drop in pressure in the condition 
A S B 3 , (Fig. 35), will equal the rise in pressure for A 1 B V 

Consider the kinetic energy of the moving column in A Q B . 
Denote this by E. This is stored as potential energy at A 1 B 1 . This 




F 

Fig. 40. — Block Model for Time History of Excess 
Pressure at any Point in Line. Ideal Case. 



is again transformed into kinetic energy at A 2 B 2 , and the latter 
will therefore equal E in amount though reversed in direction. 
This same energy E is again rendered potential at A Z B Z against 
the forces of gravity, which produce the normal condition of pres- 
sure at A B or A 2 B 2 . This means a reduction of the total potential 
energy of compression by the amount E. Relative to the total 
normal potential energy of compression at A B or A 2 B 2 , the con- 
dition at A X B X implies therefore an excess measured by E and the 



WATER RAM OR SHOCK IN WATER CONDUITS 89 

condition at A 3 B 3 a decrease likewise measured by E. This will 
obviously imply equal changes of pressure above and below the 
normal. 

It must be understood that this simple relation can only hold 
under the condition that the reduction of pressure is not greater 
than the original steady motion pressure. A liquid will not admit 
of the development of a tension, and after the pressure is reduced 
to any further tendency in the column A s B 3i for example, would 
imply a break and the entry of discontinuous conditions. 

Thus near the upper end of a penstock line where, for example, 
the steady motion pressure is 301b. absolute, if a sudden shut 
down produces first an increment of pressure measured by 50 lb., 
then the pressure at the given point will rise to 801b. abs., but 
cannot drop below abs. There will, however, in such case be 
developed a tendency for the water to separate and leave the 
upper end of the pipe producing discontinuity and turbulence as 
the manifestation of the remainder of the energy which cannot 
be absorbed by the development of a tensional stress. 



26. Modifications in Physical Conditions Necessary 
to Allow for Elasticity of Pipe, for Friction 
and for the head due to velocity 

So far as the more complete physical picture is concerned, we 
must consider that the water and the pipe form together an elastic 
system, and that the arrest of the former will result in a com- 
pression of the water and an extension or dilatation of the pipe. 
Likewise with reduction of pressure the water will expand and 
the pipe contract. The water and the pipe therefore enter con- 
jointly into all interchanges and transformations of energy between 
the kinetic and the potential forms. With these facts in mind the 
changes in the physical picture necessary to allow for the elasticity 
of the pipe will be readily made. The mathematical development 
will be found in Sec. 27. 

Regarding the pressure history, as affected by friction, let AB 
(Fig. 41) denote the line, NN the static level and KL the hydraulic 
grade for steady flow. Then, as we have seen in Sec. 14, the distances 
from AB to KL denote the values of the pressure head at points 
along the line. 

If now any portion of the line, as a differential element in length, 
is brought suddenly to rest, the kinetic energy will be suddenly 
transformed into potential energy of compression which will 
manifest itself as an excess pressure, additive to the pressure 
already in evidence at the given point. 

But the velocity is uniform throughout the length of the line 
{AB uniform in section) and hence the kinetic energy per element 
of length is uniform, Hence at each point in the line the elementary 



90 



HYDKAULICS OF PIPE LINES 



accession of energy in the compression form, due to sudden arrest, 
will be uniform and the corresponding pressure will be represented 
by some height such as LD. 

The line CD parallel to KL will therefore mark the limits of 
pressure reached at successive points in the line at the instant of 
arrest of the corresponding elements of flow, or otherwise at the 
instant when the wave of compression, starting from B, reaches the 
point in question. 

If, then, the pressure condition thus realized would remain in 
statu quo during the traverse of the compression wave front to the 
upper end of the line A, we should have, at the end of the com- 
pression phase, the entire line at rest, in a state of compression and 
with a pressure gradient CD. 

This excess pressure head gradient CD, however, is not a gradient 
of equilibrium, and in consequence the pressure conditions realized 




Fig. 41. — Excess Pressure as Modified by Effects due 
to Friction. Partial Account. 



at the instant of arrest will, during the remainder of the com- 
pression phase, be subject to further change. 

Thus suppose for the moment that with the wave at E and say 
half the line under compression, the excess pressure head gradient 
were FD. 

Then it may be readily seen that the water in the line between E 
and B, at rest and under gravity, will not be in equilibrium under 
a total pressure head distributed according to the gradient FD. 
There will be a tendency for pressure energy to flow from high to 
low or from F toward D and thus to seek a level of uniform head. 

It is of interest to note that the problem of energy flow thus 
arising is very similar to that presented by the flow of heat along a 
bar under corresponding specified temperature conditions. 

The initial value of the pressure head is always on the line CD so 
that at the end of the compression phase the pressure gradient 
must end at C. The remainder of the gradient will, however, differ 



WATER RAM OR SHOCK IN WATER CONDUITS 91 

from CD as a result of energy propagation, the part toward D being 
raised and flattened in the approach toward a uniform level, while 
that nearer C will fall below CD as a result of losses of energy 
exceeding gains. At the instant when the compression wave front 
reaches A we shall have then, the original kinetic energy repre- 
sented under the following items : 

(a) Compression energy manifested as pressure and distributed 

according to the instantaneous condition resulting from its 
generation and propagation as noted above. 

(b) A residual amount of energy in the kinetic form and involved 

in the movements connected with energy propagation. 
That is, so long as the total head is not uniform, so long 
will there be energy propagation involving molecular move- 
ment and hence kinetic energy. 

This propagation of compression energy and approach toward 
uniform distribution will, of course, continue during the return of 
the wave up to the point when the front of the wave of recovery 
reaches the point in question ; that is during the entire compressive 
phase. The extent of the unloading or recovery will furthermore 
depend, at each point, entirely upon the difference between the 
pressure necessary for the unloaded phase and the pressure in the 
compression phase at that particular point. 

But the pressure in the unloaded phase (6, Fig. 36) will depend 
in a complex manner on the influence due to friction in the establish- 
ment of the reverse flow (3 to 6, Fig. 36). Thus in Fig. 41 the 
pressure head at A or AK must remain unchanged whether the flow 
is direct or reverse. Hence with a reverse flow established we shall 
have a disappearance of the excess pressure head CD with a flow 
head represented by some gradient line such as MK . 

Now the kinetic energy of reverse flow is simply the expression in 
kinetic form, of the excess compression energy. Hence at each point 
the velocity generated will be determined by the compression 
energy in excess and thus available for transformation into the 
kinetic form. But this will vary from point to point as determined 
by the difference between the excess pressure head at the given 
point and the pressure head required to maintain the flow against 
friction. To a first approximation, these varying amounts of 
compression energy would be represented by the intercepts between 
CD and KM. There will be, therefore, a tendency to develop, at 
each point in the line AB, a different velocity, v the initial value at 
A and gradually less and less as the available compression energy is 
less due to the growing demands for friction. But continuous flow 
means uniform velocity, and hence any tendency to develop varying 
velocities along the line will in itself tend to set up a secondary 
series of waves which will themselves be subject to propagation in 
the usual manner. 

The net result of this complicated series of actions and reactions 



92 



HYDRAULICS OF PIPE LINES 



will be, at the end of the unloading phase (6, Fig. 36) a distribution 
of the total energy available under the forms : 

(a) Kinetic energy of motion in the reverse direction or from 
B to A, and representing a resultant or group velocity 
somewhat less than the initial velocity v. 
Compression energy, due to the existing complex state of 
motion, distributed along the line and unavailable for 
expression in the kinetic form. 



(6) 



E 

X—» 




F K 
G 


J 


L 


A 


D 


H 


I 


B 



Fig. 42. — Excess Pressure as Modified by Effects 
due to Friction. Partial Account. 

(c) .Compression energy as required, transformed into the work 
done against friction in setting up the reverse flow, and the 
expression of which will be a reverse flow pressure gradient, 
something like KM. 

The time history of the pressure at any given point will evidently 
partake of all these complexities of pressure generation and energy 



M 



H 



K 



AT 



Fig. 43. — Excess Pressure as Modified by Effects 
due to Friction. Partial Account. 



propagation. Aside from secondary effects due to energy propaga- 
tion, the time history would be somewhat as in Figs. 42, 43, 44. 
Using these diagrams as a first approximation we may gain some 
general idea of the character of history to be expected. Thus in 
Fig. 42 referring to conditions at the valve, let XX denote the 
static level, AB the level for velocity head and CD the running level, 
including friction head. Then pressure would start from D and on 
arrest of movement rise to E a distance DE measuring the excess 
pressure due to arrest. The further history would then consist of 
alternate levels EF, HI, etc., equidistant from XX and denoting the 
levels reached at the successive instants when the energy is all in the 



~ WATER MM OR SHOCK IN WATER CONDUITS 93 

potential form (3, 9, Fig. 36). With propagation, however, the 
line EF will actually rise according to some complex law and the 
return wave will reach not quite down to H, following which will 
be a slight down slope resulting again from energy propagation. 
The history would thus show, for the extreme pressures, a series 
of lines slightly sloping away from XX and at distances from XX 
gradually decreasing with time. 

Similarly, to a first approximation, at points in the line as 
indicated in Figs. 43, 44, the levels EF and I J, expressing the 
pressure when all in the potential form (3 or 9, Fig. 36) would lie 
equidistant from the static level XX, while the levels OH and KL 
expressing the pressure when the energy is kinetic would He at equal 
distances above and below the velocity head fine AB. 

In comparison with this incomplete ideal, the actual history 
would show sloping lines for EF and IJ, at gradually decreasing 



E 


,*.. 




i 




\ 


A , | 

c ; 

i 


D 






■ 




— i B' 

i 



Fig. 44. — Excess Pressure as Modified by Effects 
due to Friction. Partial Account. 

distances from XX and lines GH and KL nearly horizontal and at 
nearly equal distances from a velocity head fine AB which itself 
will gradually approach XX. 

It is thus seen that as the ultimate result of the influence due 
to friction, an exceedingly complex condition develops and the 
discussion of this particular phase of the problem in further detail 
is quite beyond the scope of the present work. The magnitude of 
these secondary elements can scarcely be determined by direct 
mathematical means, and the results derived by such direct methods 
must always be understood as representing a first approximation 
to actual conditions, more and more nearly accurate as the friction 
head is less and less significant as a factor in the problem. 



27. Velocity of Propagation of Acoustic Wave 

The well-known formulae of physics give for the velocity of an 
acoustic wave in an elastic medium the equation : 

s=y?--- « 

where S= velocity (fs). 
gr=gravity. 

l£=coef . of elasticity (pf2). 
w= density (pf3). 



94 



HYDRAULICS OF PIPE LINES 



In the case of water in a pipe line, however, we do not have the 
simple case of a single medium. We must include the influence of 
the steel shell forming the envelope of the water, and the problem 
becomes therefore that of finding the velocity of the wave in a core 
of elastic water surrounded by an elastic metal shell. 

Let us in the first place consider the latter. In the case of an 
excess internal pressure q it may be subject to excess stress in two 
directions, longitudinal and circumferential, measured as follows : 






(2) 



where T ± = stress in longitudinal direction (pi2). 

T 2 = stress in circumferential direction (pi2). 
q= excess pressure (pi2). 
r=radius (i). 
t— thickness (i). 

Poisson's investigations show that in the case of a plate stressed 
in two directions at right angles, the relative stretches (strains) are 
expressed in the form : 

dx_T x T 2 - 
*~ E aE .(3) 

y E aE J 

where x and y denote the original lengths in the two directions, dx 
and dy the corresponding extensions, T x and T 2 the corresponding 
stresses, E the coefficient of elasticity and a a factor known as 
Poisson's coefficient. 

Applying to the present case we may take x longitudinal and y 
circumferential. Then dyjy=d(27rr)l27rr=drlr, and from the above 
formula we have 

dx_ qr qr 

lc~2iE~"alE ^ (4) 

dr __ qr qr 

T~lE~ 2atE 

Again let K denote the cubical coefficient of elasticity for water. 
Then by definition of the meaning of this term we have 

dV_q^ 
V~K 

where V— original volume. 

dV= diminution of volume. 
q= excess pressure. 

Then dV=lv. 



(5) 



WATER RAM. OR SHOCK IN WATER CONDUITS 95 

Applying this to the original volume of water in a length of pipe 
x we have 

dV (water) = ^recompression). 

Again for the pipe, as a result of the excess pressure q and the 
resulting excess stresses, longitudinal and circumferential, the new 
length and new radius will become x-\-dx and r-\-dr respectively. 
The new volume will then be 

(V+dV)=(x+dx)>n(r+dr) 2 . 
The original volume was 

V=x nr 2 . 
Subtracting we find, after neglecting differential terms of the 
second and third orders, 

dV (pipe) =2 -nrxdr + i:r 2 dx (expansion). 
Substituting for the values of dr and dx from (4) we find after 
simple reduction of form : 

dF(pipe)=7rr 2 z/ J^\ / 5-- \ (expansion) (6) 

The total relative change of volume between the water and the 
pipe will then be measured by the compression of the former plus 
the expansion of the latter. 

Calling dV such total relative change we have 



But jcr 2 x= V the original volume. 



(7) 



>-y (8) 



V K^2tE 



We must now consider that the elastic compression of the water 
plus the elastic extension of the shell combine to give to the system 
a virtual coefficient of elasticity which we may denote by J. We 
may otherwise consider this change dV as an apparent or virtual 
change in volume, as evidenced by the shortening up of the total 
column of water, a part of such shortening being due to compression 
of the water and a part to the extension of the pipe. In any case we 
relate such virtual change in volume to a virtual coefficient of 
elasticity J for the system composed of water and pipe. 

We shall then have by definition, as in (5) : 

f=i <»> 

or from (8) q_ q qr 
J~~K + 2tE 

0T H+2^( 5 - 4 a) < 10 > 



96 HYDRAULICS OF PIPE LINES 

For steel plates we may take a=3-6, giving : 
1_1 1-944, 
J K^ tE { } 

We shall then hava forjthe combination of water and pipe, and 
similar to equation (1), 



■~J' 



gJ 



«=v» (12) 

For numerical values in equation (11) we may conveniently take 
the pound and foot as units, and hence K and E will be measured 
in pounds per square foot. 

This gives #=43,200,000. 

i£=:4,032,000,000. 

The resulting value of J substituted in (12) will then determine 
the velocity of propagation of the acoustic wave along the pipe 
line. If the pipe line were absolutely rigid, the value of J would 
equal K and the velocity S would be that for water alone, or about 
4700 fs. Due to the influence of the elastic shell, J is always less 
than K, and S is less than the value for water. Values of S for 
various values of rjt are given in Table XXIV. 





TABLE XXIV 


q per 


r/t 


S(i8) 


unit v (pi2) 


10 


4293 


57-84 


15 


4119 


55-50 


20 


3964 


53-41 


25 


3827 


51-56 


30 


3702 


49-88 


35 


3588 


48-35 


40 


3485 


46-96 


45 


3390 


45-68 


50 


3302 


44-49 


55 


3221 


43-40 


60 


3146 


42-39 


65 


3076 


41-45 


70 


3010 


40-56 


75 


2948 


39-72 


80 


2890 


38-94 


85 


2835 


38-20 


90 


2783 


37-50 


95 


2734 


36-84 


100 


2688 


36-22 


105 


2643 


35-61 


110 


2600 


35-03 


115 


2561 


34-51 


120 


2522 


33-98 



WATER RAM OR SHOCK IN WATER CONDUITS 97 







q per 


r/t 


S (fa) 


unit v (pi2) 


125 


2485 


33-48 


130 


2450 


33-01 


135 


2417 


32-56 


140 


2384 


3212 


145 


2353 


31-70 


150 


2323 


31-20 



Iron or Steel Pipe. Values of velocity of acoustic wave S and of 
pressure q developed per unit of velocity quenched. 

(r/£=ratio of radius to thickness.) 



28. Excess Pressure Developed 

We may now proceed to determine the excess pressure in the 
pipe as a result of complete and instantaneous closure of the valve. 

The kinetic energy of a cubic foot of water moving with velocity 
v is 

When the water comes momentarily to rest in the condition 1 

(Fig. 35), this energy must exist in potential form represented by 

the work done in compressing the water and in extending the pipe. 

As we have seen above, this will be equivalent to the production 

of a change of volume dV in a system of virtual cubical coefficient 

of elasticity J. 

Then as in (5) : 

dV = q_ 

V~J 

JdV 
or q=- y - 

The value of q during such compression will vary from to the 
full value, and the mean will be one-half the above or JdV/2V. 
The work done will be measured by mean pressure X change in 
volume. Hence 

w q J(dV) 2 q 2 V nq , 



Hence we shall have for one cubic foot of water 

wv 2 _ q 2 
~2g~~2J 



whence 



; <"> 



H.P.L. 



98 HYDKAULICS OF PIPE LINES 

Combining (14) with (12) we find 

ff-^-'Gfi) d5) 

or *=jg?fei2) (16) 

Let h denote the head corresponding to pressure q. Then we have 
fc=?=^0(f) (17) 

The ratio S/g recurs so frequently in the further discussion of 

these problems that it will be convenient to represent it by a single 

symbol. To this end put 

S 

- — a. 

g 

With this notation (17) becomes 

h=av (18) 

By a different mode of combining (14) and (12) so as to eliminate 
wig we also find v i 

' q=f(vm (19) 

or <7=j^(pi2) (20) 

or H^(f) (21) 

Since S is commonly found between 3000 and 4000, it follows that 
the value of q will commonly range from 40 to 54 (pi2), or the value 
of h from about 90 to 125 feet, per foot second of velocity arrested 
by instantaneous closure. 

Where the pipe line is of varying diameter and of varying thick- 
ness of metal, the theoretical investigation in any precise manner 
becomes too complicated to serve practical purposes. Average 
values may, however, be usually taken in such manner as to serve 
practical requirements. 

29. Water Ram with Rapid Complete Closure 

We have thus far assumed the closure complete and instan- 
taneous. Suppose next that it occupies a certain time interval T, 
and that the time history of the retardation produced at the valve 
is given by the curve of Fig. 45. 

Any ordinate AC of this curve is then a measure of the retarda- 
tion or rate of velocity change produced at the corresponding 
instant t. Also the area OAC will be proportional to the total 
change in velocity during time t, and the total area OAB will be 
likewise proportional to the total velocity v , which, at the end of 
time T 3 is reduced to 0, Under these conditions there will be 



WATER RAM OR SHOCK IN WATER CONDUITS 99 

initiated at the lower end of the pipe a continuous series of pressure 
waves, each representing an element of velocity change and each 
added to the one preceding, thus gradually building up the pressure 
at the valve as the sum of these elements, each of which will be 
proportional to an ordinate of the curve OAB. That is, the excess 
pressure at the valve will continuously increase by the addition 
of successive elements, each due to a retardation acting through 
an element of time and producing an element of velocity change. 
At any instant during the period of closure the total pressure at 
the valve will then be the result of a summation of all these ele- 
ments, developed from the beginning of the movement up to the 







—mm If? 


A 


D \. 


i^l 


, 


C tl J# 



Fig. 45. — Time History of dv/dt and v. 



given instant of time, and this will correspond to the total change 
in velocity produced, that is, to an area such as OAC. 

These facts stated in a physical sense may be established analyt- 
ically as follows : 

We have first to find the element of pressure at the valve due 
to a retardation dv/dt continuing through an element of time dt. 
The result is, of course, a reduction of the velocity v by the amount 
dv in time dt. 

The distance covered by the wave in time dt will be Sdt. That 
is, a length of water column Sdt will, in time dt, be retarded by the 
amount of velocity change dv. Let A— cross section area and 
w=density. Then the mass subject to this retardation is wASdtjg. 

The force required is measured by the product of mass by re- 
tardation. Let dQ denote such force. Then we have 

Jr . wA dv a - ,_ rt . 

dQ= — - — Sdt (22) 

g dt 

This will appear as an excess pressure at the valve, distributed 
over the cross section area of the column. Let dq denote the corre- 
sponding unit pressure. Then 

dq=™%Sdt (23) 

g dt 

W 

or dq=- Sdv (24) 

or again, dh=—=- dv—adv (25) 

b w g v ' 



100 HYDRAULICS OF PIPE LINES 

In (24) or (25) the only variable term is the velocity change dv. 
This equation shows that the element of excess pressure dq is 
determined by the constant factors characteristic of the case and 
by the velocity change dv. More specifically it shows that it is 
independent of the time element dt. This somewhat surprising 
result is due, as is shown by the form of equation (23), to the fact 
that the expression for the pressure element dq contains two variable 
factors, one proportional to the acceleration and the other to the 
quantity of water involved. The first of these, expressed by dvjdt 
carries the time element in the denominator, while the other, 
represented by Sdt, carries the time element in the numerator. 
In the product the time element disappears, leaving the pressure 
element dependent solely on the velocity change dv. Or otherwise 
if the rate of valve closure is increased, for instance, the retardation 
will also be increased, but the time occupied and the quantity of 
water involved will be correspondingly decreased. Thus if the 
closure is effected in one -half the time the acceleration will be 
doubled and the quantity of water halved, and thus the product 
will remain the same. 

The effect produced by each successive value of the retardation, 
as indicated by a curve such as OAB, will be entirely similar in form, 
each proportional to the element of velocity change dv, and there- 
fore for the entire curve we shall have the sum of a series 
of elements, each similar in form to (24). 

Summing these we have 

wSAv w . tul 

Q=— — =- S{v -v)=aw(v -v) (26) 

y y 
and h=aAv=a(v — v) (27) 

The expression {v — v) or Av for the velocity change occurs so 
frequently in the further discussion of these problems that we shall 
find it a convenience to represent it by a single term. To this end put 

(v -v)=s (28) 

In this sense s always means the aggregate change in velocity 
starting from the initial velocity v . 
With this notation (27) becomes 

h=as (29) 

Equation (27) is general and applies either to partial or complete 
closure. In the latter case s=v and h=av as in (18). 

It thus appears that the excess pressure at the valve will be 
dependent solely on the change of velocity produced and indepen- 
dent of the time required to realize such change. 

It is also noted, by comparison with (18), that this value of h is 
the same as for instantaneous closure. 

The independence of the excess pressure on the time of valve 
movement is, however, only realized within suitable limits as to 



WATER RAM OR SHOCK IN WATER CONDUITS 101 

length of line and time of valve movement, as will be shown at a 
later point. 

Returning now to the various pressure waves formed at the valve, 
it is seen that they will all travel with the velocity S, adding them- 
selves to the previous value of the pressure, so that when the wave 
corresponding to any ordinate (as AC, Fig. 45) has reached any 
point in the line, the pressure at such point will equal that at the 
valve when the given wave started. 

Let the broken line ODEF denote the integral curve of OAB. 
That is, a curve such that the ordinate at any point, as CD, is 
proportional to the area OAC and similarly for all other points. 
Then any ordinate as CD will represent the total velocity change 
from the origin up to time t, while the ordinate BF will represent 
the total velocity v . This curve, measuring from OB, will then give 
the time history of the velocity change ; while measuring from 
GF to the curve, we shall have the time history of the velocity itself 
as it gradually falls from v to 0. 

But as we have seen, the pressure at the valve is proportional 
to the total change in velocity and hence to the ordinates of the 
curve ODEF. In other words, this curve will give a time history 
of the growth in excess pressure at the valve. 

Again, due to the wave propagation, as noted above, it will be 
clear that at any instant t, the first wave (corresponding to £=0 
at the valve), will have reached a distance x=St, and the pressure 
condition will correspond thereto. The wave corresponding to 
the ordinate AC will just be leaving the valve, and the pressure 
there will be represented by DC. At intermediate points along the 
line, between the valve and the point x=St, the pressure condition 
will be represented by the successive ordinates of the curve OD. 
In other words, OD is a space history of the pressure distribution 
along the line between the given point and the valve. 

Similarly the entire curve ODEF will give the time history of 
the excess pressure at the valve, progressing with the time from 
at the beginning of the movement to BF at the close, and covering 
the total time of valve movement T. Likewise the same curve will 
give for this instant (at the end of T) the space distribution or history 
of the pressure along the line, represented by BO, the valve being 
at B with the pressure BF, and the farthest point reached, distant 
x=ST, being at with pressure zero. 

It follows further that at any point x=St the time history of the 
pressure will be the same as at the valve, but retarded by a time 
interval t. 

These various conclusions assume, of course, no disturbance in 
the conditions, due to reflection from the upper end of the line. 



102 HYDRAULICS OF PIPE LINES 



30. Water Ram with Rapid Partial Closure 

For simplicity of treatment, the discussion thus far has assumed 
complete closure. 

The treatment of Sec. 29 embodies, however, the more general 
case as shown by equations (26), (27). 

In the various formulae for complete closure we have in any case 
only to substitute for v , the velocity of flow, the change in velocity 
expressed by (v Q —v) = Av=s. 

Thus for any change from v Q to v, whether instantaneous or 
gradual (so long as the conditions of Sec. 31 are fulfilled) the values 
of q and h are as given in (26), (27). 

The generality of these results traces back to (24), (25) which will 
have the same form whether the closure is partial or complete. In 
any case the value of the integral of dv is Av=s, the velocity arrested, 
complete or partial as the case may be. 

Hence in one case this integral will have the value v and in the 
other (v —v) = Av=s. 

,31. Conditions for Realization of Assumptions of 

Secs. 25-30 

Reference has been made to the conditions under which the 
results of Secs. 25-30 may be realized. From the preceding discus- 
sion it will be clear that at the valve the pressure will continue to 
increase according to a time history as given by some curve such as 
ODEF (Fig. 45), unless interfered with by reflection from the upper 
end of the line as discussed in Sec. 25. Let 2>=length of line and 
T=timQ of valve movement. Then if L=ST it is clear that the first 
impulse to leave the valve will just reach the upper end of the line at 
the end of the period, and the entire history of the growth in 
pressure at the valve will be spread out along the length of the pipe. 
If the length L is less than ST the first impulses to leave the valve 
will have started back, as a reverse or partial unloading of the 
pressure, before the close of the period T. If L is only slightly less 
than ST the upper end of the line only will be affected by this 
unloading and the lower end, and especially at the valve, will show 
the same condition as in the case when L—ST. If L=ST/2 or 
T—2L/S the return from the upper end will just reach the valve at 
the close of the period. For a shorter value of L or a longer value of 
T there will be a certain amount of unloading and reflection at the 
valve, depending on the circumstances of the case. 

Broadly speaking, then, the values : 

L=ST/2 
or T=2L\S 
furnish the critical conditions regarding the pressure at the valve. 
For convenience of notation let us represent the time 2L/S by the 



WATEE RAM OR SHOCK IN WATER CONDUITS 103 

symbol z. Then if T is less than z the conditions assumed in 
Sees. 29, 30 will be realized at the valve and the pressure will be the 
same as for instantaneous valve movement. If T is greater than z 
there will be a certain amount of unloading and reflection at the 
valve and formulae (18), (29) no longer apply. 

If T lies between z/2 and z there will be a length of the line 
measured from the upper end, which will undergo partial unloading 
before the close of the valve movement — a length less and less as 
T approaches z/2. 

The condition of pressure resulting from values of T greater than 
z/2 and involving reflection back and forth from the ends of the line 
will be considered in later paragraphs. 



32. Derivation by Different Methods of Certain of 
the Preceding Formulae for Excess Pressure 

The subject of shock or water hammer in pipe lines is of so great 
general importance that it may be well to derive certain of the 
preceding results in a somewhat different manner. 

Consider first instantaneous full closure. Then holding in mind 
the total column of water of length L we find, after it has been 
brought to rest in the compressed state 1 (Fig. 35), that it has 
shortened up a certain amount. This is due, as we have seen, 
partly to the actual compression of the water and partly to the 
extension of the pipe. 

The shortening up of the column is measured by the distance 
traversed in time t=LjS by what may be termed the upper end of 
the water column with cross section A moving with velocity v . 
The distance moved is VqL/S and the total apparent change in 
volume is AvqL/S. 

If then J represents the virtual cubical coefficient of elasticity as 
defined in (9) we have as before : 

41JL 
v ~J 

AV 
or f=yJ (30) 

But AV is the total apparent decrease in volume noted above. 
Hence, 

But ^4 £= total volume = V and hence 

V s 

and from (30) vJ . , " 
q— -%- as m (19). 



104 HYDRAULICS OF PIPE LINES 

But from the law of propagation for an acoustic wave we have 
from (12) 

9 

Whence we find wSv n . ,_ „,. 

q— "=aM asin (15). 

Again assume partial instantaneous closure reducing the velocity 
from v to v v Then during the time t required for the compressive 
wave to travel the length of the line L, we shall have a discharge at 
the lower end under velocity v ± and a volume discharged AvJ. 
Likewise at the upper end we must consider the water flowing in 
under a velocity v until the lapse of time t=L/S when the com- 
pressive wave will reach the upper end and the entire column for 
the moment will be in a state of compression and moving with 
velocity v v The total volume of inflow will then be measured by 
Av t. The difference between the inflow at the upper end and out- 
flow at the lower will measure the apparent change in volume due to 
water compression and pipe extension. Hence we shall have 

AV= ^o-Vt)L (31) 

We have again the fundamental relation : 

~V~J 

AV T 
or q=-y J. 

But AL= V and from (31) we have 

V b 

which is the generalized form of (19). 

Again substituting for J as in (12) we find 

q= ^ ^ = aws, as m (26). 

Again consider the work energy relation. The volume of the pipe 
will be AL and the original energy wALv 2 /2g. After time LjS this 
energy will be represented by the following items : 

1. The kinetic energy of the water in the pipe moving with 

velocity v x and measured by wALv-^j2g. 

2. The work done in forcing out at the lower end the volume of 

water Av^/8 under the excess pressure q. This will be 
measured by qAv-JjjS. 



WATER RAM OR SHOCK IN WATER CONDUITS 105 

3. The work done in producing the apparent change of volume 
A Y—A (Vq—v^L/S. This will be measured by 
qA(v — v-^L/28. See equation (13). 
We may then write a work energy equation : 
wALv 2 



2g 

Combining and reducing we find 



)+(2)+(3). 

as in (26). 



9 

33. Water Ram in Pipe Lines When Lower End of 
Pipe is Held Rigid 

The formulae and methods of Sees. 25-30 assume that the lower 
end of the pipe is free to move longitudinally. Only on this assump- 
tion can longitudinal stress be developed in accordance with 
equation (2) and only on this assumption will the virtual coefficient 
of elasticity J have the value as given in equation (10). 

If we now assume the lower end of the pipe rigidly fixed, then 
dx=0 and we have as the only change in the dimensions of the pipe 

dr_pr 
7~tE 
Following this value through with exactly the same method as in 
Sec. 27, we find 

dV _p_p 2pr 

1 1 , 2r 
° r J=K+tE 

Comparing this with (11) we find it the same except for the 
coefficient 2 as compared with 1*944. 

Referring to equation (3), it is seen that the presence of a stress 
(and its resulting strain) at right angles to a given stress will reduce 
the strain or extension which the latter stress would by itself 
produce. The circumferential stretch due to a given internal 
pressure will then be less than if the circumferential stress existed 
alone and not in conjunction with the longitudinal stress. With the 
latter eliminated, as in the case of a pipe with the end rigid, the 
circumferential stretch will therefore be larger than with the two 
stresses coexisting and, as the equation shows, the total extension in 
volume is practically the same in the one case as in the other. In 
fact so far as the equations apply and with the value of Poisson's 
modulus assumed for steel, it appears that the volume expansion of 
the pipe is slightly greater with the lower end rigid than when free. 
Practically the difference is not significant. 

We may therefore conclude that whether the lower end of the 



106 HYDRAULICS OF PIPE LINES 

pipe is rigid or free the total volume expansion under an excess 
pressure p will be substantially the same, and therefore the equations 
of Sees. 27-30 for J and for the excess pressure resulting from a 
sudden quenching of velocity may be practically employed indepen- 
dent of the degree of constraint of the lower end of the pipe. 



34. Rapid Opening from Complete Closure 

In the case of a rapid or practically instantaneous opening of the 
valve from complete closure, the case comes under the general 
method of treatment of Sees. 25-30. A similar system of acoustic 
waves will be formed, beginning with a wave of expansion or drop of 
pressure, and followed by excess and defect in alternation, as 
discussed in Sec. 25. The amplitude of these pressure waves will, 
however, not be the same as for the case of sudden closure. If the 
opening is instantaneous, the pressure at the valve will be reduced 
from the total static head behind the valve to the pressure on the 
discharge side. 

Actually the valve cannot be opened instantaneously and the 
drop in pressure will be somewhat less than this amount. Let q 
denote the pressure drop and v the velocity of the water movement in 
the wave toward the valve. Then there will subsist between q and v 
for pressure drop and velocity generated the same relation as in 
Sec. 28 for pressure rise and velocity quenched, and we shall have 
for the velocity of the water forming the wave toward the valve 
the value, 



v=q Jhr\ ' see (14) (18) (32) 



Where h now denotes the head due to pressure drop. 

This wave will be propagated toward the upper end of the line 
with a velocity S as in Sec. 25. In case the gradient of the pipe is 
such that at all points the absolute statical pressure is greater 
than q, then during the passage of the wave the pressure will at all 
points remain positive and the wave will reach the upper end with its 
amplitude or pressure drop practically unchanged. At this instant 
the entire column of water is moving toward the valve with the 
velocity v and with a pressure q below the normal value. At the 
inlet, however, the pressure must remain normal and reflection at 
this point will result in the propagation toward the valve of a wave 
of normal pressure and with a water velocity v relative to the 
expanded part or 2v relative to the pipe. This will in turn be 
reflected at the valve under conditions representing a change in 
velocity between 2v and the existing velocity at the valve. If there 
are no disturbing conditions, and especially if effective reflection 
from the partly open valve could be realized, then the result would 
be an excess pressure wave q propagated up the line ; and thus the 
series of pressure drop and pressure excess would alternate, forming 



WATER RAM OR SHOCK IN WATER CONDUITS 107 

a general pressure history similar to the case for closure as discussed 
in Sec. 25. 

In the actual case, however, disturbing conditions may enter, and 
in any event the reflection from a partly open valve is imperfect. 
The conditions contemplated in the physical picture are not there- 
fore completely realized, and the excess pressure is usually con- 
siderably less than the pressure drop and the series of alternations 
of pressure above and below normal rapidly damps out to a negligible 
amount. 

Again, if due to the gradient of the line there should be a point 
where the pressure drop would render the pressure zero or negative, 
then there will result a discontinuity in the physical conditions of the 
problem and the wave will travel on to the inlet with a greatly 
reduced amplitude. The reflected wave will then represent a much 
reduced velocity and the series of pressure fluctuations will dampen 
out very rapidly. 

Under certain conditions of discontinuity with a rapidly opened 
valve, the return pressure may not even pass the normal static value, 
or indeed it may not even reach such value, the return from the 
initial pressure drop gradually flattening out to the pressure value 
with friction head under final steady conditions. 

Broadly speaking, the initial pressure drop will vary directly with 
the degree of initial opening. With full opening from closure the 
drop will reach nearly down to the pressure on the discharge side of 
the valve. With only partial opening, the initial drop will be 
reduced. 

The further analytical development with discussion of this case 
will be found in Sees. 41, 42, as a special case of the general problem 
of valve opening. 

35. Rapid Opening from Partial Initial Opening 

The general phenomena attendant on such cases are broadly 
similar to the case of opening from initial closure, with a closer 
and closer approach as the initial opening is less. With increase in 
the initial opening there is a rapid decrease in the initial pressure 
drop and a decrease in the value of the return excess pressure 
wave, and a rapid approach toward the condition of dead beat 
return from the initial drop to the final steady flow pressure value. 

The analytical treatment of this case will be found in Sec. 41 
as a special case of the general problem of valve opening. 



36. Law of Increase of Pressure with Time, Valve 

Closure 

The equations developed in Sees. 28-30 give the maximum or 
ultimate value of the pressure reached, but do not furnish any 
indication of the time history of the growth of such pressure. 



108 HYDRAULICS OF PIPE LINES 

These equations apply at the valve so long as the time of valve 
movement T is not longer than z=2L/S, and throughout the pipe 
so long as T is not greater than zfe—L/S. 

The excess pressure will clearly depend, among other things, 
on the rate of closure of the valve. The usual assumption in this 
connection is of uniform closure ; that is, of a uniform rate of 
decrease of valve opening. 

Independent of any such assumption regarding the rate of valve 
closure, however, but assuming T not greater than 2LJS we may 
investigate the law of pressure change as follows : 

Let A = cross section area of pipe . 

a — area of valve opening. 

#=velocity of acoustic wave. 

u= velocity through valve. 
v =original value of v. 

/= coefficient of efflux through valve. 

h=qlw— excess pressure head. 
Put m=a/A. 

We have then three equations as follows : 

v=mu (33) 

This expresses the continuity of flow along the pipe and through 
the valve. 

h=a(v —v) = as (34) 

This expresses, in accordance with (27), the excess head developed 
at the valve corresponding to any reduction of velocity (v —v), and 
hence the excess head at the valve at the instant when the pipe line 
velocity is v. 

u 2 / Lv 2 \* 

|=/( ff +*-fc) < 35 > 

This expresses the value of the head on the discharge side of 
the valve, u 2 /2g, transformed under efficiency / from the total net 
head on the upper side of the valve, made up of the original head H 
plus the excess pressure head h, minus the friction head Lv 2 /C 2 r 
(see Chap. I (44)). 

Putting (35) all in terms of u and transforming we have 

. Mu 2 =H+h (36) 

* In this expression, the term a y (see Chap. I (44)), representing the differ- 
ence in the external pressure at the two ends of the line, is omitted. This 
does not involve any lack of generality in the present treatment. In any of 
the subsequent equations of the present chapter, wherever H occurs, H -f a y 
may be substituted for it, if Ay has a value other than 0, thus giving full 
generality of treatment in this respect. 



WATER RAM OR SHOCK IN WATER CONDUITS 109 

If then between (33), (34) and (36) we eliminate u and v and 
reduce the equation in h we have 

k>-2 ^ 0+ (|^y +(a ^-2^g 2 =0 (37) 

Put E = av 

F= {am) 2 

2M 

Then (37) takes the form 

h 2 -2(E+F)h+E 2 -2FH=0 (38) 

Solving this as a quadratic in h we have 

h=(E+F)-^F*+2F{H+E) (39) 

When m=ra we find E 2 =2FH, and h in (39) reduces to h=0 
as it should. 

When m—0, F=0 and h reduces to av as it should. 

If then we take a series of values of m from m to and corre- 
sponding to valve movement from start to full closure, we may, 
with the known hydraulic characteristics of the case, find the values 
of E and F and hence of h. 

We may then find s from (34), thence v and u if desired from (33). 

The time history of the pressure head h and the velocity v is 
shown for four typical cases in Fig. 48. 

In a the value of h remains negligibly small during the early part 
of the movement, only beginning to rise at the very last, and then 
jumping with great rapidity to its maximum value at the instant 
of complete closure. 

In b the rise is slow at first and then more rapid, but in less pro- 
nounced degree than in the case of a. 

In c the curve is of the same general character, but more nearly 
approaches a uniform rate of pressure rise. 

In d the curve is only slightly convex to the axis of time, showing 
a nearly uniform rate of increase. 

The history of «s will be the same as that of h and the history 
of v will hence be similar, but in the inverse direction, as noted on 
the diagrams. 

In a the reduction in velocity is negligible during the valve move- 
ment up to the last tenth and then the velocity is rapidly reduced 
to zero, accompanied by the rapid upshoot in the value of h as 
noted above. In b, c, the rate of reduction of v changes progres- 
sively toward the condition indicated in d, where a nearly uniform 
rate of reduction is realized. 

The characteristics of these various cases, as noted, lead to the 
following general conclusions : 

The controlling condition giving rise to a pressure history such 
as that of a is a large value of ra , that is a valve or nozzle opening 
nearly or quite the full size of pipe. In the case of a, m =l-00. 
This implies a pipe with gravity flow and hence the entire head H 



110 HYDRAULICS OF PIPE LINES 

used up in friction and velocity, or since the latter head is small, 
it follows that substantially the entire head H is used up in 
friction. 

Other things equal, the head In will be small with large values 
of F, and will increase more and more slowly with decrease in m 
in accordance as the value of L is greater and greater. 

Going to the other extreme as represented by the case of d the 
initial value of m is very small (-02) and the head is very high 
(2700 (f )). At full pipe line velocity of 8-04 (fs) the velocity head 
is about 1 foot and the friction head is 85-5. Hence but a small 
fraction (3-2%) of the total head is used up in friction and velocity, 
the remainder (96-8%) being available under steady conditions as 
pressure head. 

The cases represented in b and c are intermediate between those 
of a and d. The fraction of head used in friction in the four cases 
is progressively -968, -65, -17, -032. While other factors will modify 
the form of the curve to some extent, the general progression from 
that of a to that of d will correspond to a lesser and lesser fraction 
of the total head absorbed in friction. 



37. Law of Decrease of Pressure with Time, 
Valve Opening 

The treatment of the case of valve opening is in effect contained 
within that of closure, as in Sec. 36. It is only necessary to remember 
that for valve opening, h is negative and to modify the resulting 
equations accordingly. 

The treatment of this case in further detail will, however, be 
found in Sec. 41, as a part of the more general treatment of the 
case of valve opening. 

38. Gradual Closure: Time Long Relative to Time 
2L/S for Double Traverse of Acoustic Wave* 

In this case there will be a series of reflections back and forth 
from the two ends of the line, somewhat after the manner assumed 
in Sec. 25. The total effect, however, may be considered as made 
up as a summation of successive effects due to successive move- 
ments of the valve and to the consequent successive elements of 
velocity reduction and the resulting elements of pressure change. 
Thus as in Sec. 29 an elementary or differential change in velocity 

* Notation : In this and following sections of the present chapter it will 
be found of special convenience to denote the value of the reduction in 
velocity (v -v) by the single symbol s and also a series of values of h, s and 
other quantities belonging to a series of time instants t, tr-z, t-2z, ir-3z, etc., 
by h, h lf h 2 , h s , etc. That is, no subscript implies a value relating to the 
instant of time t, a subscript 1, to an instant earlier by z, a subscript 2, to an 
instant earlier by 2z, etc. 



WATEK RAM OR SHOCK IN WATER CONDUITS 111 

dv taking place in an elementary or differential time dt will 
produce at the valve an elementary or differential pressure change 
measured by ^^ 

and this element will move with velocity S as an elementary pres- 
sure wave along the pipe and suffer reflection back and forth some- 
what as indicated in Sec. 25. 

At any point in the line therefore, and at any subsequent time, 
the total net excess pressure head h will be the algebraic summation 
of all such elements, both direct and reflected, as have, during such 
time, reached or affected such point. 

With reference to such conclusion, however, one important 
reservation must be made. 

The reflection of pressure waves in a liquid back and forth from 
the two ends of the line, as assumed in Sec. 25, assumes complete 
or perfect reflection from each end of the line. Also in the ideal case, 
the damping effects due to viscosity are neglected. 

With regard to reflection from the upper end of the line, such 
reflection is based on the condition of a constant pressure at this 
point, and such condition must obviously be fulfilled, since just 
beyond the upper and open end of the pipe we can only have the 
reservoir pressure, which is assumed constant in value. Hence 
we may with propriety assume that, at this end of the line, reflec- 
tion will be realized with a close approach in manner and degree 
as assumed. 

On the other hand, the reflection from the valve end assumes 
the valve closed before the reflected wave returns to this end of 
the line. A dead end, and at which the velocity must become zero, 
is therefore the implied condition for the complete reflection assumed 
in these earlier sections. With the conditions of the present section, 
however, the reflected wave returns to the valve end before closure 
and while water is still issuing. The reflection cannot therefore be 
complete. 

The degree to which reflection is realized will presumably depend 
on the closeness to which the condition of a dead end at the valve 
with zero pipe line velocity is approached. 

Thus with an area of valve opening at the start the full size of 
the pipe and in the early stages of the movement, the area will 
be but slightly reduced, the excess pressure developed will tend to 
increase the issuing velocity u, and there will be but slight reduction 
in the main pipe velocity v, and in consequence the reflection must 
be quite incomplete. As the valve approaches the closed position, 
however, the area through the valve will become much reduced, the 
pipe velocity v will become small and more complete reflection 
should be realized. Otherwise, considering the valve as equivalent 
to a diaphragm moved across the opening, we may say that in the 
early stages of the movement there is but a small area of diaphragm 
available against which a reflected wave can form, while near the 



112 HYDKAULICS OF PIPE LINES 

close of the movement such area will be larger, and more complete 
reflection will be realized. 

Again, in the case of a pipe line under high head and with a 
nozzle area a, even when wide open, small compared with the pipe 
line area A, it seems probable that, at all stages of the valve move- 
ment, relatively more complete reflection should be realized. The 
closure of a nozzle from area a to zero may, in effect, be considered 
as the last stages of a closure of the complete area from A down 
through a and to zero. From another viewpoint we may consider 
that the end area available for the support of a reflected pressure 
wave will be (A— a) even when the valve is wide open and will 
gradually increase to A as the valve is closed. 

These points still remain in uncertainty, however, and there is 
much need for further experimental study of the general problem 
in order to determine more definitely the extent to which reflection 
of pressure waves can be realized from the delivery end of a pipe 
discharging water through a nozzle or opening of varying area in 
relation to the c.s. area of the pipe. 

We must, however, in general, conclude that the reflection oi 
pressure waves from the valve end of a pipe under discharge will 
be more or less imperfect or incomplete, approaching complete 
reflection as the flow of water becomes less and less, and realizing 
substantially complete reflection from and after the moment of 
valve closure. 

In Sec. 40 will be found some further discussion of the subject of 
partial reflection at the valve. 

The condition of complete reflection may in general be con- 
sidered as a limiting case to which actual cases will approach as the 
attendant circumstances may determine. It becomes therefore a 
matter of interest to develop, at least in general outline, the results 
which may be expected in such limiting case. 

To this end and holding in mind the principles of Sec. 25 we have 

dh=a(dv—2dv 1 +2dv t —2dv t +etc.) (40) 

whence 

h] t o =a(v] t o -2v]^+2v] t o '-2v]^-{-etc.) (41) 

Or with the special notation of the present section, 

h=a(s-2s 1 +2s 2 -2s 3 +etc.) (42) 

Thus in (40) the first term represents the element generated at the 
valve at the instant t, the second term the element generated at the 
valve at the instant (t—z) or t x and which has, in the meantime, 
travelled to the upper end of the fine and back again, arriving at the 
instant t and, by complete reflection as discussed in Sec. 25, operates 
to reduce the pressure condition at that instant by 2adv t - z , or by 
twice the amount of the element generated at the instant {t—z). 
Similarly the third term represents the element generated at the 
valve at the instant (t—2z) and which has in the meantime com- 
pleted one full cycle of four traverses of the length L, and thus 



WATER RAM OR SHOCK IN WATER CONDUITS 113 

returns to the valve at the instant t as a wave of positive pressure 
and by reflection then gives a positive element of 2adv t _ 2 z or twice 
the element generated at the instant (t—2z). 

The remaining terms, however numerous, indicate successive 
elements generated at the valve at successive time intervals of z 
counting backward from the instant under consideration. The 
number of terms, therefore, will be given by the whole number next 
below tjz. These terms, as readily seen, will have alternately minus 
and plus signs according as they have made an odd or an even 
number of double traverses of the length L. 

In equation (41) the successive terms indicate each the summa- 
tion of a series of elements adv, all of which are similar in sense and 
time history (number of double traverses of pipe line length). Thus 
for the first term the time period is to t, giving the summed effect 
(all in the positive sense) of all elements as formed and previous to 
propagation or reflection. For the second term the time interval 
is to (t—z) or t lt giving the summed effect (all in the negative sense) 
of all elements which have had time to make the double traverse 2L 
with return to the valve and reflection at that point. For the third 
term the time interval is to (t—2z) or t 2 , giving the summed effect 
(all in the positive sense) of all elements which have had time to 
make two double traverses, or one complete cycle, with return to 
the valve in the positive sense and reflection at that point. 

In this manner the various terms are made up, each representing 
the summation of elements which have made at least a given whole 
number of round trips from the valve to the upper end and return, 
successively 0, 1, 2, 3, etc. 

It should be noted that this entire development of an expression 
for the resultant h in the case of a closure extending over a time t 
larger than z is only an extension, by the process of summation, of 
the principles and methods developed in Sees. 25 and 29. 

Re -writing (42) we have 

h=a(s—2s 1 +2s 2i — 2<s 3 +etc.) (42) 

We shall have similarly 

h 1 =a{s 1 — 2s 2 +2s 3 — etc.) (43) 

It is readily seen that these two expressions after the terms in 5 X 
have the same terms with opposite signs. Hence 

h^-h 1 =a(s-s 1 ) (44) 

Again, in equation (42) put 

B=2s 1 — 2s 2 +2s 3 — etc (45) 

We have then h=a{s-B) (46) 

Noting the make-up of h ±i as in (43), we also readily see that 
/i 1 =a(-B— s x ) 
or aB=h 1 -{-as 1 (47) 

Thus from either equations (44) or (46), (47) it appears that the 
value of h for a given time t can be immediately determined if we 

H.P.L. — I 



114 HYDKAULICS OF PIPE LINES 

can find s for the same time and knowing also h and s for a time 
(t—z). Or otherwise if we know h and s for a time t and can find s 
for a time t-\-zwe can find h for t-\-z and so on for a series of values 
of t separated by the interval z. 

Those results are of remarkable simplicity, connecting, as they do, 
successive values of h separated by the time interval z. 

The determination of the values of h during any period of time in 
general, involves three distinct phases or time periods : 

1 . t between o and z. 

2. t between z and T. 

3. t beyond T. 

With proper interpretation (44), (46) and (47) apply generally 
to all three periods. Thus for the first period the subscript „1 
implies a time (t— z) negative and in such case the term is to be 
omitted, giving in (44) : h==as ( 48 ) 

For the second period the equations apply as written. 
For the third period, for t=T and beyond, s becomes v . Hence 
for values of t between T and T-\-z we shall have from (44) : 

H^ah-sJ (49) 

while for (t— z)>T or t>(T-\-z), both s and s x become v , and we 

have h=-h x (50) 

It now remains to provide means for the determination of the 
value of s corresponding to any given time t. 

In Sec. 36 equations have been developed for the determination 
of the time history of each of the various quantities h or q, u, v, s, 
and on the assumption of a time of closure T less than z. These 
equations will therefore apply directly and without change to the 
present problem for the first time period from £=0 to t=z. 

For the second time period from t=z to closure, or t=T, we 
proceed as follows : 

Equations (33) and (36) apply to this time period without change. 
Instead of (34) we have (46), and this becomes 

h=a(v -v-B) (51) 

Combining these so as to eliminate u and v, as in Sec. 36, and 
reducing the equation in h we have similar to (39) : 

h=(E+F)-<^F*+2F (H+E) (52) 

Where E=av —aB 

jF= W asbefore '* 

and 0^=^-1-05! 
When m=0, F=0 and h=E=av —aB as it should from (51) 
with v=0. 

* For an expression for F not explicitly involving the quantities rn and / 
(as in notation of equation 36), see Appendix II. 



WATER RAM OR SHOCK IN WATER CONDUITS 115 

It will be noted that (52) differs from (39) only in the term E 
which here appears as av —aB instead of av . 
Repeating for convenience (33) and (36) we have 

v=mu (53) 

Mu 2 =H+h (54) 

and m—a/A and where a may vary in any specified manner with 
the time. 

The condition of uniform rate of valve area closure is often 
assumed in dealing with problems of shock. In such case if a is the 
original or full opening we shall have 

-= a iH) =m °H) (55) 

It should be especially noted, however, that in this general 
method of treatment it is not necessary that the valve follow any 
particular law in closing and in particular that the treatment is in 
no wise limited to the case of linear closure. The various equations 
of the present section expressing h as the sum of a series of terms 
are entirely independent of any term expressing the time rate of 
valve movement. The valve opening is represented solely by the 
factor m which appears in equations (36) and (38), and no matter 
what the character of the valve movement may be, so long as it is 
known, it will be possible to assign to a series of values of t the 
corresponding series of values of m. This insures, therefore, the 
solution of the problem for any assigned rate or character of valve 
movement. 

In any case, therefore, the program for finding a series of values 
of h extending over any period of time from to t is as follows : 

1. The conditions of the valve movement determine the values 
of m. If the ratio of closure is assumed uniform then m for any 
value of t is given by (55). 

2. From the known hydraulic and other characteristics of the 
case find the values of the various terms in equation (52), omitting B. 

3. With a time interval At as selected and the various numerical 
values of the terms in (52), putting B—Q, find the corresponding 
series of values of h up to the value of t next smaller than z. Then 
find the resulting values of s, v and u, if desired, using (48) and (53). 
This will give a series of values covering the period up to z or up to 
the nearest point below z. A further point may then be found in 
the same manner for t—z. If the time interval is exactly contained 
in z, the regular series will contain the point for t=z as its last 
member. A value of the time interval At which is either equal to z 
or an even submultiple is to be advised in a program of computation 
of this character. 



116 HYDEAULICS OF PIPE LINES 

4. Then taking for t a value z-\-At, find from (47) the value of 
aB=h 1 -\-as 1 =2s 1 or 2s at time At, and with this value of B and the 
appropriate values of the other terms find from (52) the value of h, 
and thence s from (46) and thence v and u if desired. 

5. Continue the process by taking next t=z-\-2 At and find from 
(47) the value of B=2s 1 or 2s at time 2 At, and similarly for successive 
steps. Beyond t=2z the value of B will no longer equal 2s ± but will 
always be correctly given by (47). 

6. Beyond t= T the procedure is simplified through (49), (50) and, 
as will be seen, the time history of h then becomes a periodic curve 
with alternate positive and negative maxima equal to the value 
realized at t=T, and with a complete period of 2z. 

It thus appears, in the development of such a series of values 
of h, that each set of terms found for a time t serves to determine 
through (47) and (52) the h and thence the s, v and u for the 
time t-\-z. 

It further results that a value of h at any single value of t cannot 
in general be found except as a member of the series extending at 
least from t—z. The complex condition of multiple reflection, 
back and forth, does not seem to permit of representation in a single 
equation with known terms, unless, indeed, such equation were made 
of such complexity as to represent in effect the series of operations 
required to determine the series of values as above indicated. 

Sample Computation. — In order to indicate the character of the 
numerical operations in connection with the solution of equation 
(52) the following sample computation is given.* 

The basic data taken are as follows : 

£=4000- (f). 
D= I- (f). 
H= 500- (f). 
ChezyCoef. G= 110- 

/= -96. 
£=4000- (fs), 
m = *05. 

Arrest of valve at half closure or ra^-025. 

Rate of valve closure -10m =-005 in time z=2 (s). 

We then find L/C 2 r=h322 and l/2^/=-01618. 

The initial conditions are found in the column under £=-00. To 
this end only the quantities necessary are entered. M=l/2gf-\- 
Lm 2 /C 2 r results as shown, and then u 2 =H-^rM and v=mu. 

For t=z, B=0 and hence E=av Q . 

M is next found from the value of m=-045 and similarly F. 

In this form of computation equation (52) is put in the form 
(56), and the solution proceeds as indicated, giving a value &=44-8. 
The remainder of the column then furnishes a check on the numerical 
accuracy of the work, and also serves to determine E for the next 
step with t—2z. 

* See also Sec. 39. 



WATER RAM OR SHOCK IN WATER CONDUITS 117 



{H-\-Ji)^rM (see equation 
-v) and aB n from aB n = 



Thus we find u from the relation u 2 = 
(54)), and v from v=mu, s from s = (v - 
(h+as) (see equation (47)). 

In this notation B n denotes the B for the next step ahead. 

Then with these values we find E n from E n =(av —B n ), where 
likewise E n denotes the value of E for the next step ahead. This 
gives E n =905-5, which is then entered as the E for the next step 
t=2z, and thus the computation proceeds. 

The intermediate steps for t=3z and 4z are omitted and the 
results then given for t=5z carried through in the same manner. 

Then for t=Qz, m and hence M and F remain the same as for 
t=5z, and thus the computation follows through giving a value 
of A=- 16-67. 

The numerical check referred to is found in the relation h=a(s—B) 
(see equation (46)). Thus the difference between the as in one column 
and the aB n found in the preceding column should equal the value 
of h. It will be noted that this relation checks out within the 
nearest tenth, which is as close as the number of places included in 
the computation will secure. 

The full results for the case up to t=10z are shown graphically 
in Fig. 566. 



SAMPLE COMPUTATION 



1. 


t 


•00 


z 


2z 


5z 


6z 


2. 


m 


•050 


•045 


•040 


•025 


•025 


3. 


E 




995- 


905-5 


612-9 


507- 


4. 


m 2 


•0025 


•0020 


•0016 


•00063 




5. 


1 -322m 2 


•0033 


•00268 


•00211 


•00082 




6. 


•01618 


•01618 


•01618 


•01618 


•01618 




7. 


M 


•01948 


•01886 


•01829 


•01700 


•01700 


8. 


am 




5-59 


4-969 


3 106 




9. 


(am) 2 




31-25 


24-69 


9-647 




10. 


2M 




•03772 


•03658 




•03400 


11. 


F' 




828-6 


675- 


283-7 


283-7 


12. 


H + E 




1495- 


1405-5 


1112-9 


1007- 


13. 


H+E+F 




2323-6 


2080-5 


1396-6 


1290-7 


14. 


{H + E) 2 


2235025- 1975400- 1238600- 1014050 


15. 


(H + E-\-F)* 


5399100- 4328400- 1950500- 1665900- 


16. 


Diff. 


3164075- 2353000- 711900- 651850- 


17. 


V~ 




1778-8 


1534- 


843-75 


807-37 


18. 


(E + F) 




1823-6 


1580-5 


896-60 


790-70 


19. 


h 


•00 


44-8 


46-5 


52-85 ( 


-)16-67 


20. 


H+h 


500- 


544-8 


546-5 


552-85 


483-33 


21. 


+M 


25665- 


28890- : 


29880- ! 


52520- 28430- 


22. 


u=V 


160-2 


170- 


172-86 


180-33 


168-61 


23. 


V 


801 


7-65 


6-914 


4-508 


4-215 


24. 


v o 


801 


8 01 


8-01 


801 


8-01 


25. 




•00 


•36 


1096 


3-502 


3-795 


26. 


as 


•00 


44-7 


1361 


435-1 


471-4 


27. 


aB n 


•00 


89-5 


182-6 


487-95 


454-73 


28. 


av 


995- 


995- 


995- 


995- 


995- 


29. 


En 


995- 


905-5 


812-4 


507- 


540-27 



118 



HYDRAULICS OF PIPE LINES 



Graphical Treatment of Equation (52).— It is seen that equa- 
tion (52) may be readily put into the form 

h=(E+F)-^/(i±+E+F)*-(H+E)* (56) 

This form readily lends itself to graphical solution as in Fig. 46. 
Lay off AB, BC, CD representing respectively H, E and F. 
Then AD=H+E+F. With AG as radius draw the arc CQ. To 
this draw the tangent DP and then the radius A P. Then AP— 
H+E and PD =the radical in (56) and swinging over the arc 
BE from D as centre we have h=PB. 

Pressure at any Point in the Line. — The discussion thus far has 
related solely to the pressure at the valve. The same principles 




Fig. 46. — Graphical Solution of Equation giving Value 
of Pressure Head h. 



as developed in Sec. 38 and generalized for any point P at a distance 
x from the valve, enable us to write down a general equation similar 
to (42) as follows : 

ft=a(s«_i— ^.(g.i)— s*_(*4.i)+s*-(2*-i)-f-S/-(2*+i)— etc.)*. . .(57) 

As in (43) each of these terms represents the summation of those 
parts of the final result which have had, so to speak, a common 
life and which thus admit of summation. The first term is the 
summation of the elements generated at the valve, and which have 
at least traversed the distance x between the valve and the point 
P, distant from the valve by the time interval *. The second term 
is the summation of the elements generated at the valve, and 
which have at least traversed the distance L to the upper end and 
back again to the point P, a distance 2L—x and requiring a time 
(z—i). These come back as an unloading of pressure and hence 

* In this and subsequent equations, i=time x/S. 



WATER RAM OR SHOCK IN WATER CONDUITS 119 

appear with the negative sign. The third term is similarly the sum- 
mation of the elements which have at least traversed the distance 
L to the upper end and back again to the valve and back by reflec- 
tion there to the point P, a distance 2L-\-x and requiring a time 
(z-\-i). This likewise operates as an unloading term and hence 
appears with the negative sign. Similarly the other terms repre- 
sent the summation of elements corresponding each to a common 
distance of propagation and approaching P alternately from the 
upper end and from the valve. 

It is of interest to note that when the point P is at the valve and 
x=0, the two unloadings represented by the second and third 
terms occur simultaneously at the valve, and become by their 
sum 2s t _ z or 2s x as in equation (42). In this manner by the assump- 
tion of x=0 or ^=0, equation (57) becomes reduced to (42), and 
the latter is thus seen to be only a special case of the former when 
x=0. 

Writing equation (57) for time (t—z) and adding the two values 
thus found, we have by the cancellation of all terms after the 
second, h+h 1 =a(8 t _ i -s tHz -4 ) ) (58) 

It will be noted that at any given instant of time, s is the same 
throughout the length of the pipe. Hence the investigation 
of the pressure at the valve end will give a series of values of s 
which may be applied throughout the length of the line as indi- 
cated in (57) or (58). 

To this end we need a general history of s on time, and this may 
be most conveniently laid down graphically from the results derived 
for the valve end. 

With such a graphical history of s, the determination of h for 
any point in the line and at any time becomes, through (57) and 
(58) a matter of simple routine. 

Equations (57) and (58) are entirely general and may be thus 
applied over any time period, with the single interpretation that 
when any subscript is zero or negative, such term is to be omitted. 
In this manner (57) could be employed as far as might be desired. 
If the number of terms becomes large, however, the operation 
grows in numerical complexity, and in this case the relation ex- 
pressed in (58) may be employed with advantage. 

It is also of interest to note in (57) that if x=L, i=z/2 and the 
successive terms become equal in pairs with opposite signs and 
thus h=0 as it should. 

Assumption of Uniform Value of dvjdt. — In discussing the problem 
of shock the assumption is sometimes made of a uniform value of 
dvjdt, that is, of a uniform retardation in the velocity v. In such 
case each of the elements dv will be equal, and the value of s up 
to t—T will always be given by 

s=nt (59) 

Where n is the constant value of— dvjdt 



120 



HYDRAULICS OF PIPE LINES 



Equation (44) for this case becomes 

h-\-h 1 —anz= constant (60) 

An examination of (42), remembering that subscripts when or 
negative imply the omission of the term, shows readily that h 
starting from increases uniformly up to t—z, where it reaches the 
value anz. It then begins to decrease at the same rate, as is also 
implied in (60). Consideration of (42) and (60) shows that in such 
case the graphical history of h would consist of a series of straight 
line slopes running from to a maximum of anz and then down 
to again, and with a complete time period of 2z. This will hold 
as long as the valve is in movement. After closure and beyond 
t=T-\-z, we shall have, as before noted, h=—h v That is, any two 
values with a time interval z will be equal and apposite in sign. 
Examination of this part of the history in detail will show that the 
graph will be as indicated in Fig. 47, where the full lines show the 
up and down slopes during the period of valve movement, and the 
dotted lines show the various forms which the graph would take 




Fia. 47. — History of Pressure Head h assuming Uniform Rate 
of Velocity Change (Closure). 



after closure, dependant on the value at t=T. Thus if T<z the 
graph will follow the dotted line as indicated from i on. If the 
terminal point when t=T lies on a down slope as at C the graph 
will follow the dotted line as indicated and similarly for other 
cases. 

These graphs are readily seen to fulfil the following laws : 

1. Values separated by a time interval z are equal and apposite 

in sign. 

2. The complete period is 2z. 

3. The graphs will in general have flat tops and sloping sides, 

the slope of the sides being twice that of the fuU line part 
belonging to the period of valve movement. 

4. The flat tops mark maximum and minimum values, which 

are alternately positive and negative in sign and equal in 
numerical value. 

5. The numerical value of the maximum (or minimum) equals the 

value reached at the instant of valve closure. 



WATER RAM OR SHOCK IN WATER CONDUITS 121 

6. If the instant of valve closure is on an up slope, as at A or D, 

the graph starts off on a flat top at this value of h and 
continues for the rest of a period z. It then falls until 
t=T-\-z, and reaches a numerically equal negative value, 
and so on as shown. 

7. If the instant of valve closure is on a down slope as at (7, 

the graph falls during the remainder of a period z, reaching 
a negative value equal numerically to the value at valve 
closure, and then starts off with a constant value of h, which 
it holds until t=T-\-z, when it rises and follows its course 
as shown. 

It is of interest to note that if the instant of valve closure occurs 
at the end of an odd number of periods z, bringing the point to the 
top of one of the slopes, the graph will continue as a series of right 
line slopes implying a series of positive and negative extreme 
values equal to cmz. 

If, on the other hand, the instant of valve closure occurs at the 
end of an even number of periods z, bringing the point to the bottom 
of one of the slopes where h=0, the value of h will continue to hold 
zero as a constant value, and the excess pressure head in this case 
disappears with the closure of the valve and remains indefinitely 
thereafter. 

While this analysis of the conditions resulting from an assumed 
uniform value of dvjdt has an interest as a part of the general 
problem, and especially as illustrating the effective manner in which 
the equations above deduced serve to determine a wide diversity 
of results according to changing conditions, it must be recognized 
that the fundamental condition of uniform retardation is one not 
likely to be met with in actual practice. 



39. Gradual Partial Closure, Time as in 
Section 38 

The discussion of Sec. 38 has assumed full closure as the final 
condition. If instead the valve remains partly open, we have in 
effect an arrest of valve movement at a time T followed by a con- 
stant value of m and such values of h, s, v, u as may develop. We 
have here to distinguish two periods : 

1. The period from £=0 to t=T, when the valve movement is 

arrested. 

2. The period subsequent to t=T. 

Obviously the condition for the first period is the same as for 
the corresponding part of a case of assumed full closure with the 
same rate of valve movement up to t=T, and as treated in 
Sec. 38. 

For the second period we shall have relations between h, s, v, u 



122 HYDKAULICS OF PIPE LINES 

expressed by the same general equations as before but now with 
a constant value of m. 

The general procedure for finding the time history of h during 
this period is therefore the same as described above for the case 
of full closure, but with a constant value of m. This will give an 
entirely different course to the sequence of values. Often the 
graph of h during this period will show a series of excursions posi- 
tive and negative of decreasing amplitude and gradually approach- 
ing 0, while the values of v and u will approach those for steady 
conditions under the given fixed value of m. The general character 
of such graphs is given in Figs. 54-56. In some cases, as determined 
by the characteristics, the return of h to the zero value may be 
by gradual decrements without oscillation through positive and 
negative values (see Fig. 54). While the equations would, further- 
more, indicate an indefinitely long period of time for the attainment 
of final steady conditions, the influence of viscosity and the failure 
to realize reflection as assumed, will rapidly obliterate the oscilla- 
tions of pressure and reduce the values to those for final steady 
flow conditions. The rapidity of return to zero, or to a negligible 
value, also varies in marked degree with the characteristics of the 
case (see Figs. 55, 56). 

Naturally, in dealing with partial closure, any law of reflection at 
the valve end may be assumed, either full reflection or partial, 
according to the various methods of estimate as discussed in 
Sec. 40. 

40. Partial Reflection at Valve (Closure) 

The treatment thus far has assumed full reflection of the pressure 
wave at the valve, a condition which, as we have already seen, will 
be only imperfectly realized in practice. It becomes, therefore, of 
interest to examine the results of an assumption of partial rather 
than full reflection at the valve. 

There are at least four assumptions which may be taken as a 
basis for the specification of a partial reflection, as follows : 

1. A constant fraction of full reflection. 

2. A fraction of full reflection measured by the ratio (m — m)/m . 

That is, reflection in proportion as the valve opening is 
reduced in area from full opening to closure. 

3. A fraction of full reflection measured by the ratio (A— a) /A. 

That is, reflection in proportion as the cross section area of 
the pipe A is closed over at the valve end. 

4. A fraction of full reflection measured by the ratio — . That 

is, reflection in proportion to the decrease in the velocity of 
the water. 
We shall briefly indicate the results as developed from these 
various assumptions regarding the degree of reflection realized. 



WATER RAM OR SHOCK IN WATER CONDUITS 123 

Case 1. — Let /denote the constant fraction. Then equation (42) 
will become 

h=a(s-(l+f)s 1 +f(l+f)s 2 -P(l+f)s 3 +etc.) (61) 

The second term represents at time t an unloading s x with reflec- 
tion fs v The third term develops first at time (t—z) as an unloading 
s 2 with reflection fs 2 and then at time Us a second unloading fs 2 
with second reflection f 2 s 2 , giving by the sum of the two latter the 
term as written ; and similarly for subsequent terms. 

Denote all terms in the parenthesis after the first by B we then 

have h=a(s-B) (62) 

With the make-up of (61) we then readily find, in a manner similar 
to that followed with full reflection the following relations : 

h+fh 1 = a(8S 1 ) (63) 

aB=fh 1 +as 1 (64) 

It will be noted that these equations all reduce to the forms for 
full reflection, as in (42), (44), (47) if we put /=1. 

We may then proceed, exactly as indicated for full reflection, 
using equations (52), (53), (54) but with the value of B as in (64). 
Case 2, — We shall in this case have for / a varying value given by 

m —m 
m 
There will be, therefore, a value of / for each instant of time 
during the closure, and in particular a series of values for the 
instants t, t—z, t—2z, etc. These we may denote by /, f v / 2 , etc., 
the same as for the series of values of h, s, etc. 
We shall have, then, instead of (61) the equation : 

A=a[«-(l+/)*i+/i(l+M-/i/2(l+/)*8+eto.]. • • .(65) 
The second term represents at time t an unloading s x plus a 
reflection fs v The third term develops first at time {t—z) as an 
unloading s 2 with reflection / 1( s 2 and then at time f as a second 
unloading f 1 s 2 with second reflection ffe^ giving by the sum of the 
two latter the term as written ; and similarly for the other terms. 

Denote, as before, everything within the brackets after the first 
term by B. Put also 

^=51— /i*2+/i/a*3— etc (66) 

Then P 1 =s 2 -f 2 s 3 -{- etc (67) 

Whence we derive 

or P=s l -f 1 P 1 (68) 

We have then for (65) h, =a (s-B) (69) 

and from the composition of (65), (66) it is seen that 

B=(l+f)P (70) 

We also readily derive the relation 

A+» 1 =a(«-J5» l -P 1 (l-J5f 1 )) (71) 



124 HYDRAULICS OF PIPE LINES 

It will be noted that (65) and (71) reduce to the forms for full 
reflection if we put /= 1 . 

We may now proceed as described for the case with full reflection 
finding h from (52) with the value of B as given by (70), (68) and 
finding s and v from (69) or (71). 

Case 3. — In this case the resulting fundamental equation is the 
same as (65) for Case 2, but, with different values of/, f lt etc., we 
shall have here : 

f=^p=l-j=(l-m) (72) 

With a prescribed program of movement for the valve, therefore, 
/ becomes known for any instant of time and we may then use the 
equations of Case 2 but with the appropriate values of /. 

Case 4. — In this case we have for valve closure 



(73) 



and hence for the series /, f lt f 2 , etc., we shall have s/v , s 1 /v , s 2 jv^ 
etc. 

We may, therefore, use the same general equations as for Case 2 
except that in this case we do not know, for any given time t, the 
value of /, since this depends on v and this in turn on u or h. This 
implicit relation does not, however, introduce any new variable 
into the equations and we proceed by taking equations (33), (36), 
(69) with the special values of B and / as given in (70) and (73). 
From these equations, and substituting for s in terms of v, we 
eliminate v, u and s, and derive, as in Sec. 36, the value of h, 

h=(E+F)-^F 2 +2F (H+E) (74) 

where E=a(v —2P) 



, „ (am) 2 . . 

and '-w' 1 






the same in form as in Sec. 36 and in the previous cases, and differ- 
ing only in the values of E and F. 

We may, therefore, follow through step by step first using (39) for 
t<z, and to determine the initial values of P, and then using these 
results in (74) for the determination of h, u, v and s for times between 
z and 22, and then using the latter values to determine those for the 
next interval z, and so on as previously indicated for the case of full 
reflection. 

As between these four bases for estimating the degree of reflection 
realized, we have little experimental evidence as a guide. There is 
no reason for assuming a constant value of f : but a constant value 
somewhat less than 1 would presumably give more accurate results 
than to assume it constant at 1, as with the assumption of complete 
reflection. 

For a case where the open valve area is the full size of the pipe, 






WATER RAM OR SHOCK IN WATER CONDUITS 125 

assumptions 2 and 3 become the same, and in such case assumption 4 
will differ widely from these in the values of / through the period of 
closure. 

For a case where the open valve area is small relative to the cross 
section area of pipe, assumption 3 will imply practically complete 
reflection, while in such case assumptions 2 and 4 will more nearly 
agree. 

41. Gradual Opening: Time Long Relative to Time 
2L/S for Double Traverse of Acoustic Wave 

The treatment for the gradual opening of a valve is, in principle, 
contained in that of Sec. 38 for closure. We shall have the same 
relations as expressed in the fundamental equations of that section 
but with the following interpretations : 

The change of head h is now a decrease instead of an increase and 
is essentially subtractive instead of additive. 

The change of velocity s will be reversed in direction ; that is we 
shaU have s =v—v 

instead of (28) and if the opening starts from complete closure we 
shall have S=:V 

With this understanding we may, parallel to (36), (51), (33), write 
the fundamental equations as follows : 
Mu 2 =H-h 

h=a(v—v Q —B) 
v=mu 
Combining these as for the case of closure, we derive the equation 
for h in the same general form 

h*+2h(E+F)+E 2 —2HF=0 

or h=-(E+F)+^/F*-t-2F(H+E) (75) 

Where E=av +aB 

2M 
and aB=h 1 J r as 1 as before. 

If the valve movement starts from full closure, we shall have 
v =0 and E=aB. 

It follows r^ctturally that this general method of treatment may 
be applied to the determination of the value of h at any point in the 
line and assuming any law of reflection at the valve end, the same 
as for closure and as developed in detail in Sec. 38. 

With the proper definition of the quantities involved and with 
the proper interpretation of the results, therefore, the entire treat- 
ment developed in Sec. 38 may be directly applied to the case of 
opening as well as closure. 

It must, however, be remembered that the total pressure head 
at any point in the line cannot become negative and hence, any 



-. 



126 HYDRAULICS OF PIPE LINES 

result implying a final absolute pressure negative in value will imply 
rather a break in the water column, turbulent conditions and a 
state of affairs generally where the physical basis assumed for the 
treatment of Sec. 38 no longer exists. 

In (75), if we put h=0 and reduce, we find 

E 2 =2FH. 

This is then the condition that h=0. If E 2 >2FH, h is negative, 
in this case implying an excess pressure. If E 2 <2FH, h is positive, 
implying here a lowering or decrease of pressure. 

In some cases F may be large compared with (H-\-E) and hence 
(H-\-E-\-F) large compared with (H-\-E). In such case an approxi- 
mate value of the radical (see equation (56)) will be (H-\-E+F)— 
{H+E) 2 /2(H+E+F). This will give 

(H+E) 2 
2(H+E+F) 

With F large and E zero or small (opening from complete closure 
or near closure) the term following H will be small and hence the 
drop in head h at the time of arrest of valve movement will be 
nearly the entire head H, leaving the net pressure head represented 
only by the small term (H+E^^iH+E+F), or in other words, 
reducing the absolute pressure head nearly to that due to the 
atmosphere. 

This condition will be approximately realized for all values of F 
equal to or greater than eight to ten times (H-\-E) or for 

#'5(8 to 10) (H-\-E) 

with a closer and closer approach to the limit h=H, as the ratio of F 
to (H-\-E) exceeds these lower values. 

The same as in the case of closure the general problem of opening 
includes three time periods : 

1 . t between and z 

2. t between z and T 

3. t beyond T. 

First Time Period, t—0 to t=z. 

For this period, B=0 and the value of E reduces to av . Further, 
if the movement is from complete closure, v =0, and we have 

h=-F+-^F 2 +2FH (76) 

This gives, then, the value of h where the valve movement is 
completed within the time z. 

It is of interest to compare this with the value of h for closure 
through the same range of. valve movement and within the same 
time. 

Let m and v denote the final steady motion values at the end of 
opening or at the beginning of closure. Then substituting for F its 
value (am) 2 ]2M and noting that #/il[/= steady motion u 2 and that 
wV=steady motion v 2 , it is readily shown that 2FH=(av) 2 —E 2 . 



WATER RAM OR SHOCK IN WATER CONDUITS 127 

Then designating the two values of h by subscripts c and o for 
closure and opening we have : 

h c =av (77) 

h o =^/F2+(av)*-F (78) 

and from these we readily derive the relation 

h c *=h *+2h F (79) 

In the general problem of valve opening we may have, for the 
initial phenomena, two cases according as T is greater or less 
than z. 

(a) T greater than z. 

In this case, from what precedes, it is clear that we shall normally 
expect a rapid drop at the start ; either reaching h=H practically, 
and holding such value up to t=z or approaching such value more or 
less closely as determined by the characteristics of the case and in 
particular by the relative values of F and (H-\-E). 

(b) T less than z. 

In this case we shall have the same general trend of values as in 
(a) except that whatever value is realized at t= T will he held uniform 
until t—z. 

Second Time Period. t=z to t=T. 

For this period the general equation (75) must be employed. The 
course of the values of the net pressure head will show a return 
upward from the low value reached at t=z followed by a course 
which will be determined by the circumstances of the case and 
which will be illustrated at a later point. 

Third Time Period. t=T onward. 

For this period the fundamental relations expressed in equation 
(75) still hold, but with the special condition m= const. This 
will simplify somewhat the operations involved, but in general this 
equation must be used in order to determine the course of the 
pressure from t—T until substantially steady conditions are 
realized. It may be noted, of course, that in case T should be less 
than z the value of h will remain constant at its value for t=T until 
t=z, after which equation (75) with m=const. will begin to apply. 

The general course of h during this third period will show, 
normally, a series of fluctuations gradually bringing the net pres- 
sure head to the value for final steady flow conditions corresponding 
to the given amount of valve opening. 

Illustrative cases will be noted at a later point. 

Partial Reflection at Valve (Opening). — The discussion of Sec. 40, 
with the equations developed, applies without change to the case 
of valve opening, except that in Case 2 the value of /will be{m 1 — m) 
/m 1 where m 1 is the ultimate value of m, while in Case 4 we shall have 



128 HYDKAULICS OF PIPE LINES 

Where v 1 =ultimate steady motion velocity with m=m^ 
E=a(v +2P) 
{am)*/ F 



42. Discussion of Formuue of Sections 38-41 with 
Numerical Cases* 

Closure. Time of Valve Movement T Equal to or Less than Time z 
for Double Traverse of Acoustic Wave. — This case is treated in 
Sec. 36. Its treatment is also, of course, contained within that of 
Sec. 38. 

The maximum value of h is always found at the instant of com- 
plete closure and is measured by 

h m =av Q . 

If the closure is only partial the maximum value of h is always 
found at the end of valve movement and is measured by 

h m =as=a(v — v). 

It will, however, require a solution of the equation (52) in order 
to determine the value of h or v. 

The course of the pressure rise in various cases has been dis- 
cussed in Sec. 36 and in connection with the diagrams of Fig. 48. 

Closure. Time of Valve Movement T Greater than Time z for 
Double Traverse of Acoustic Wave. — This case is the one most 
likely to arise in practice. A large number of numerical cases 
have been worked through and from which the diagrams of Figs. 
49-56 have been prepared. 

During the first period from t=0 to t=z, the course of the curve 
is rising, the same as for the corresponding part of one of the curves 
of Fig. 48. 

* In Figs. 48-86 showing pressure and velocity change for various cases 
of valve closure or opening, the following characteristics have been assumed, 
for convenience, uniform in value : 

Chezy coefficient = 110, and hence 
Or = 3025. 
Coefficient efflux /= -96, and hence 
l/2gf =-016177. 
Velocity S = 4000 (fs). 
Note should also be made that in the formulae relating to closure (equa- 
tion (52), etc. ) h positive implies excess of pressure, while in the formulae relating 
to opening (equation (75), etc.) h positive implies defect of pressure or depres- 
sion. Also in all diagrams relating to closure, h positive is laid off upward 
while in those relating to opening, h positive is laid off downward. 

On certain diagrams, the heavy dot calls attention to the point where 
6= z and the circle to the point of arrest of valve movement. 
Full lines refer to pressure head h. 
Dotted lines refer to velocity v. 



WATEK EAM OK SHOCK IN WATER CONDUITS 129 



1000 




7 .6 

Scale of m/m 

Fig. 48. — History of Pressure Head h and Velocity v (Closure). 

L H m m x z T 

Case a 4000 80 1-00 -00 2-0 z or less 

„ b „ 130 15 00 „ z „ 

„ c „ £500 -05 -00 „ z „ 

„ d „ 2700 -02 -00 „ h „ 





1000 




Scale 


of m 
















.150 
900 


.135 


.120 


.105 


.090 


.075 


.060 


.045 


.030 


.015 1 




800 


















f—i 




700 




''-'-^Z'. 


'■~^,^ 




***% 








/ 




600 








"""*^ 


\ 


¥<* 






// 


§ 


500 










"^ 


'X 




at 


/» 


Hi 


400 














X 


K/y 




ft 


300 
















<\ 






200 












> 


s 


\ 


s 




100 










, ^ 


^ 






NV 


















C ' 







1.0 



.9 



■« .7 .0 

Sco/e of m/m 

Fig. 49. — History of Pressure Head h and Velocity v ( Closure )* 

L H m m x z T 

Case a 4000 130 -15 20 z 

„ b „ „ -15 „ 22 

„ c „ „ -15 „ 102 

H.P.L.— K 



130 



HYDKAULICS OF PIPE LINES 



From this point on the course will depend on the time T in 
relation to z and on the hydraulic characteristics of the case. 
The more typical cases are as follows : 

(a) The curve may continue on reproducing very nearly the 
form of Fig. 48a, but reaching a value of h somewhat 
lower at the point of complete closure. 

Fig. 496 represents a case of this type. 

{b) The curve may continue on rising gradually, but with a 
more decided break at the point where t=z and reaching 
a maximum value at complete closure considerably less 
than in the case where T<z. 

Scale of rn 



705 
400 



300 



200 



100 



.04 



UJ 



DF 



10 .8 .6 J .2 JO 

Scale of m/m 
Fig. 50. — History or Pressure Head h (Closure). 

L H m m 1 z T 

20,000 850 -05 10 2z 

Dotted line shows continuation for Case T=z 



Figs. 49c, 50 represent cases of this type. 

(c) The curve may continue nearly horizontal, or with a nearly 

constant value of h, until the closure is complete. In some 
cases there may be a slight gradual rise, or again a gentle 
rise to a maximum followed by a slight decline, or again 
an immediate sharp decline. 

Figs. 51&, c, 52b represent cases of this type. 

(d) The curve may break into a series of slopes up and down, 

giving alternate maxima and minima, and implying a 
periodic fluctuation in the pressure with a total period 
of 2z. Again, the general trend of such a history may be 
gradually up or down or practically horizontal. 

Fig. 52c represents a case of this type. 

Where the head H is mostly used up in friction, as in the case of 
Figs. 48a, 49a, a moderate lengthening of the time T will give a 
result similar to that of Fig. 496. That is, the same general character 
will be retained with a sharp increase in h during the last moments 



WATER RAM OR SHOCK IN WATER CONDUITS 131 

of closure, but reaching a final value less and less as T is longer 
and longer. The important point here is that the maximum value 
is not reached until complete closure and determination through 
any form of computation on the basis of the theory of Sec. 38 must 
necessarily proceed according to the methods outlined in that 
section, and no approximate formula based on an assumed form 
of time history entirely different in character can be expected to 
give results having any rational relation with those furnished by 
the more complete theory. 

Passing to the other extreme in a case where only a very small 
part of the head H is used in friction, as in Fig. 52, and where the 





WOO 




Scale of m 


















.050 
900 


.045 


.040 


.035 


.030 


.025 


.020 


.015 


.010 


.005 / 


c 




800 


















/ 






700 


'*-^ 














/ 




7 




600 




\ 


""■■^ 








/ 


/ 






"t 


500 






\ 


- 


V, 


/ 


/ 








<■«-. 


400 










V\ 


*i^ 








4 


"a 


300 












X " x 








a 






b 




3 
2 




200 












^ v 




v 






100 


















> 


















c 




^\ 






















\ 




1 


.0 


9 


8 


7 


6 


5 


4 


1 


2 


/ 


J 



000 



Scale of m/m. 
Fig. 51. — History of Pressure Head h and Velocity v (Closure \ 



Case a S00 
„ b „ 
„ c 



430 -05 -00 -4 z 
•05 -00 „ 2zl 
•05 -00 „ 10/ 



curve for Ti^z is nearly linear, the result of increasing T will 
be to give a periodic or fluctuating form of curve as in c. For 
intermediate conditions the curve will assume intermediate forms, 
as in Figs. 50, 51. 

^jln most cases of a break-up of the curve into periodic fluctuations 
with the period 2z, the maximum value of h will not greatly exceed 
that for the instant when t=z. Hence in such cases a good approxi- 
mation to the maximum value will usually be given by solving 
equation (52) with the value of m when t=z. 

The same will be true for cases where the curve continues 
practically at a constant value of h, or with only a slight rise as 
in Fig. 516, c. 



132 



HYDRAULICS OF PIPE LINES 



For cases such as those of Figs. 49, 50, however, the value of h 
continues to rise to the end, and the value for t=z will not give a 
proper indication of the final or maximum value reached. 

A detailed analysis of equation (52) would serve to give certain 
indications regarding the course of the curve beyond the point 
where t=z, and therefore as to whether the value of h for this point 
might be taken as a reasonable maximum. Such analysis, how- 
ever, is complex and hardly more useful than the direct trial of 
one or two points. Such a trial will usually serve to give some 



100C 



Scale of m 




Fig. 52. 



J £ .5 

Scale of m/m 
-History of Pressure Head h and Velocity v (Closure). 
L H m m x z T 

Case a 20,000 3000 02 00 10 z 
.. b „ „ -02 -00 „ 2z 



•02 -00 



102 



The velocity curve for Case b differs so slightly from that for Case a that 
no attempt is made to show the two lines separately. 



indication of the future course of the curve and hence of the probable 
location of the maximum value of h. 

It will also be found, if the curve has a flat top or nearly uniform 
value of h from t—z onward, that the Allievi formula (81) or the 
mass-acceleration formula (95) will give a good approximation to 
the value. 

The main question remains as to the form of the|curve, and 
whether any such approximate formulae will apply with reasonable 
accuracy. The simplest way of answering this question will usually 
be through the actual examination of the curve itself by equation 
(52), and this will in itself give directly the values sought, if carried 
to the point of maximum value of h. 



WATER RAM OR SHOCK IN WATER CONDUITS 133 

The time required for a computation of this character will not 
usually exceed a period reasonably permissible in such a study. 
After becoming accustomed to the work it will be found that with 
the aid of slide rule and table of squares and square roots a single 
point can be comfortably determined in about ten minutes, and 
thus a curve for ^=102, or say twenty seconds for a line 4000 
feet long, in a couple of hours or less. If the solution is carried out 
graphically through the method of Fig. 46 the time will be reduced, 
though naturally with some loss of accuracy. 
400 



200 S 












/ 












200 












400 













4z 



5z 



6z 



z 2z 3z 

Time 

Fig. 53. — History op Pressure Head h showing Course after 
Fuxii Closure. 

L H m a m, z T 



800 430 -05 -00 -40 2z 




2 Time 2z 3z 4z 5z 

Fig. 54. — History of Pressure Head h after Arrest 
of Valve Movement at -6z (Closure). 

Rate of valve closure same as for complete closure in time z. 



L 


H 


m 


m x 


z 


T 


800 


42 


•20 


•08 


•40 


•6z 



Course of History of h after Arrest of Valve Movement. — After 
arrest of the valve at full closure, the curve of h, as indicated in 
equation (50), will fluctuate between plus and minus values equal to 
the value reached at the instant of closure. Such a course is indi- 
cated in Fig. 53. As previously noted, secondary disturbances 
not included in the theory will ultimately reduce the amplitude 
to a negligible amount. 

In the case of arrest of valve movement at partial opening, the 
value of h will return to zero, either by a series of stepwise slopes or 
by alternate fluctuations, as indicated in Figs. 54, 55, 56. 

Valve Opening from Complete Closure. Time T Equal to or Less 
than Time z for Double Traverse of Acoustic Wave. — In this case 



134 



HYDRAULICS OF PIPE LINES 





800 
















8 
6 

,© 
2^ 


u 


600\ 
















"2 


400 


\ / 


V 












&3 


200 / 


/ \ 


% 


\. a 


— — " 


** ~ 


- 


— 


/ 




•\ 


-,,, 




•■" / 


\ 


- — i 





// 




200. 






\ 


y 1 / 












400 






v. 


/ 































Time 



2z 



4z 



Fia. 55. — History of Pressure Head h after Arrest 

of Valve Movement (Closure). 

Rate of valve closure same as for complete closure in time z\ 





L 


H 


m 


m x 


z 


T 


Case a 


4000 


500 


•05 


•02 


20 


•6z 


» b 


>» 


j> 


•05 


•01 


>» 


•8z 




Fig. 56. — History of Pressure Head h after Arrest of 
Valve Movement (Closure). 

Rate of valve closure same as for complete closure in time lOz. 

L H m m x z T 

Case a 800 42 -20 08 -40 Gz 

„ b 4000 500 -05 025 200 5z 



WATER RAM OR SHOCK IN WATER CONDUITS 135 

there will be a continuous drop in pressure (increase in value of h), 
reaching a minimum at the close of the movement when t=T. 

Where the ultimate opening is nearly or quite the full size of 
the pipe and hence the friction head a large part of the total head 
H and with F large relative to H, the drop will be abrupt and will 
reach down nearly to atmospheric pressure. Where the ultimate 










Time 






z 













I 


















^100 





\ 

40 


\ 
















200 


■5: 


60 


X 


v 








V 






300 


c 

=5 


\a 










/ 








400 




80 








^^ 















. 


2 


I 


6 


y i 














Scale of m/m, 

Fig. 57. — History of Pressure Head h (Opening). 
L H m Q m 1 z T 
Case a 4000 80 -00 100 2 00 z 
„ tri f„ 500 -00 -05 „ z 



Time 
O z 2z 3z 4z 5z 6z 7s 8. 


c 


10 


















20 


















30 
















E5 


\40 
















-^^ 

















Fig. 58. — History of Pressure Head h after Arrest 
of Valve (Opening). 

L H m n m, z T 



800 42 -00 -20 -40 z 



opening is small relative to the full size of pipe the drop will be more 
gradual and the minimum will be higher. That is, the ultimate 
values of h will approach more and more nearly to H as the drop 
is more abrupt, in accordance with the conditions noted above, 
and contrariwise in opposite cases. These characteristics are shown 
in Fig. 57. 

Following the minimum value reached when t=T, the pressure 



136 



HYDRAULICS OF PIPE LINES 



head h will return toward zero, the course of the history depending 
on the circumstances of the case. Three typical courses may be 
noted. 

(a) The head h will return toward and ultimately to zero by 
way of a series of gentle stepwise lifts, as shown in the 
case of Fig. 58. In an actual case, due to the damping-out 
effects of friction and other secondary causes, such a case 
would develop as a practically smooth gentle up-slope 
Fgradually approaching the axis of zero pressure. 
(6) The head h will return toward and ultimately to zero by way 
of a series of more definitely marked stepwise lifts, as 
shown in the cases of Figs. 576, 59, 60a. 
(c) The head h will show a series of alternations on either side 
of the axis, as in the case of Figs. 606, 61, 62. This means 
an alternation of value, plus and minus, with diminishing 
amplitude and gradual approach to ultimate zero value. 



o 


Ti 


me 


• 


# 


3z 


5a 


100 








(.... 




■8 


]200 




( 










\300 


s 












V 




> 










400 





Fig. 59. — History of Pressure Head 
Valve (Opening). 
L H m n m, 



h after Arrest of 



20,000 425 -00 100 10 



In cases where the ultimate opening is nearly or quite the full 
size of pipe and the friction head forms the larger part of the total 
head H, while at the same time H itself is moderate or small, the 
form of the return will be similar to that of Fig. 58. In similar 
cases, but where H itself is large (implying L large), the form will 
be similar rather to those of Figs. 576, 59. 

In cases where the ultimate opening is small compared with the 
size of pipe and where the friction head forms but a small part of 
the total head H, but where H itself is large (implying L large), the 
return will be by way of alternating plus and minus values (type c), 
as in Fig. 606. In similar cases, but where H is moderate or small, 
the return may be by way of type (a) or (6), as the values involved 
may determine. 

Intermediate cases will depend on the values involved. Generally 
let m 1 be the final value m for full or ultimate opening. Then the 



WATER RAM OR SHOCK IN WATER CONDUITS 137 



smaller the value of H/m 1 the more definitely will the return be by- 
way of a curve of type (a), while the larger the value of H\m x the 
more definitely will the return be by way of a curve of type (c), 
while type (b) will be found for intermediate values. 



Time 




Fig. 60. — History of Pressure Head h after Arrest of 
Valve (Opening). 
L H m n m, z T 



Case 



Time 



4000 200 
„ 2700 



•00 
•00 



•10 20 z 

•02 „ z 




Fig. 61. — History of Pressure Head h after Arrest of 
Valve (Opening). 

L H m m x z T 

20,000 850 -00 0-5 10 z 



Actually there is no sharp line of demarcation between these 
various cases, and, geometrically, one type of curve shades into 
another by insensible gradations. 

A detailed and somewhat complex analysis of the equations 



138 



HYDRAULICS OF PIPE LINES 



involved would make possible the laying down of more definite 
criteria regarding the type of curve to be expected in any particular 
case. This, however, does not seem justifiable in the present work. 
Maximum value of h. In the present case, as noted above, the 
maximum value of h is found at the end of the valve movement or 
when t=T. It may therefore be computed from equation (75) by 
substitution of the proper values. 



^ 800 

ft) 



Time 2z 



3z 



4z 



400 














/ 


400\ 






/ 


800 >^ 









Fig. 62. — History of Pressure Head h after Arrest of 
Valve (Opening). 

L H m m 1 z T 

20,000 3000 00 02 10 z 



/Ti 




z Time 2z 3z 4z 5z 6 


lb 


























"^ 












f' 


1 
&0 


r- 


v_ 






f* 








r 


80 


r c 



Fig. 63. — History of Pressure Head h after Arrest of 
Valve (Opening). 

L H m e m x z T 
4000 80 -00 100 2-0 2z 



Valve Opening from Complete Closure. Time T Greater than 
Time z for Double Traverse of Acoustic Wave. — In this case we may 
note the three periods as before : 

1. From £=0 to t=z the pressure will drop continuously follow- 

ing, for that part of the opening up to t=z, the same law as 
in the case oiT=z. At the instant t=z the maximum value 
of h or greatest drop in pressure will be reached. 

2. From t=z to t=T the curve will rise either in periodic lifts 

or following a series of alternations about a mean value 



WATER RAM OR SHOCK IN WATER CONDUITS 139 



Time 
2z 3z 4z 5z 6z 7z 8z 9z lO z 



20 



<40 



80 



100 

I Fig. 64. — History of Pressure Head h with Increase in 
|Value of T (Opening). 

L H m mx z T 
4000 80 00 100 2 lOz 



z Time 2z 




200 
Fig. 65. — History of Pressure Head h (Opening) T>z. 



L 


H 


m n 


m 1 


z T 


4000 


200 


•00 


•10 


2-0 22 



Time 



400 



2z 3z 4z 5z 6z 7z 8z 9z IQ z 




Fig. 66. — History of Pressure Head h (Opening). 
Influence of Increasing T. 





L 


H 


m n 


Why 


z 


!T 


Case a \ 


800 


430 


•00 


•05 


•40 


z 


„ b 


»» 


„ 


•00 


•05 


>> 


2z 


„ c 


>> 


>> 


•00 


•05 


>> 


5z 


» d 


»» 


»> 


•00 


•05 


r »» 


10* 



140 HYDRAULICS OF PIPE LINES 

holding a general trend nearly horizontal or slightly up- 
ward. Here again a small value of U\m x will normally 
determine a form similar to that of Fig. 58 or in Fig. 63 
for the time period z to 2z, or again in Fig. 64. On the 
other hand a large value will, for the same time of opening, 



10Z 




Fig. 67. — History of Pressure Head h (Opening). 
Influence of Increasing T. 



Case 





L 


H 


m 


m 1 


z 


T 


a 


20,000 


850 


•00 


•05 


10- 


z 


b 


„ 


„ 


•00 


•05 


>5 


2z 


c 


>> 


»> 


•00 


•05 


„ 


Sz 


d 


>> 


,, 


•00 


•05 


>> 


4z 



lOz 




200 



Fig. 68. — History of Pressure Head h (Opening) 
Influence of Increasing T. 





L 


H 


m 


m 1 


z 


T 


Case a 


4000 


200 


■00 


•10 


20 


z 


„ b 


>> 


>> 


•00 


•10 


jj 


2z 


„ c 


J5 




•00 


•10 




5z 


„ d 


»> 




•00 


•10 


J3 


lOz 



determine forms showing marked periodic lifts or alternations 
about a mean value, either nearly uniform or slightly rising 
(see Figs. 65-70). As shown by the diagrams these cases 
shade insensibly the one into another. 
3. For t beyond T, the value of h will return to either by periodic 
lifts or by alternate plus and minus values similar to those 



WATER RAM OR SHOCK IN WATER CONDUITS 141 



O z 2z \z 4z 5z 6z 7z & 9z lO z 




100 



Fig. 69. — History of Pressure Head h (Opening). 
L H m m x z T$ 
4000 500 -00 -05 2 lOz 



Time 
2z 3z 



4z 



5z 



6z 



7z 



8z 



9z lOz 



50\ 


















^o 


100 \ 












\ 


X 







Fig. 70. — History op Pressure Head h (Opening). 

L H m m x z T 

4000 2700 00 02 2 lOz 



* 20 

■s 40 

^60 



Scale of m/m, 
.2 .4 



^6 



.8 



1.0 









d 


- {) 




s^e 








\b 










I 








i) 



80 

Fig. 71. — History op Pressure Head h, starting prom 
Partial Opening. 

Rate of Valve Opening the same in all cases. 

L U m m x z T 

Case a 4000 80 -00 100 2 z 

„ b „ „ -05 1-00 „ -95z 

„ c „ „ -10 1-00 „ -90z 

„ d , -50 1-00 „ -50z 



142 



HYDRAULICS OF PIPE LINES 



of Figs. 57-62, for return after T=z. In general the return 
is without alternation of sign except for large values of 
the ratio Hjm combined with values of T only moderately 
greater than z (see also Figs. 63, 65). 
Maximum value of h. In this case the maximum value of h 

Scale of m/m, 
2 J .6 .8 1. 




Fig. 72. — History op Pressure Head h, starting from 
Partial Opening. 

Rate of Valve Opening the same in all cases, 

L H m lm x z T 

Case a f 4000 500 -00 ^-05 2 z 

„ b "* „ „ -01 -05 „ -8z 

„ c „ „ -025 -05 „ -52 

„ d „ „ -04 -05 „ 2z 

Time 2z 




Fig. 73. — History op Pressure Head h, 
Half Opening. 



starting prom 



L H m 

Case a 4000 200 -05 

„ b „ „ -05 



m x z T 
•10 20 z 
•10 „ 5z 



(drop in pressure) is found when t—z. It may therefore be com- 
puted from equation (75) by substitution of the proper values. 

Valve Opening from Initial Partial Opening. Time T Equal to 
or Less than Time z for Double Traverse of Acoustic Wave. — In cases 
where the ultimate valve opening is nearly or quite the full size 



WATEK RAM OR SHOCK IN WATER CONDUITS 143 

of pipe and the initial opening one-half the ultimate or greater, the 
initial velocity v Q will be but slightly less than the final velocity v v 
In such case the increase in velocity from v to v x will be small, and 
the whole program of values for h will be correspondingly reduced. 




Fig. 74 — History of Pressure Head h, starting from 
Half Opening. 

L H m m x z T 

Case a 800 430 -025 -05 -40 z 

„ b „ „ -025 -05 „ 52! 








z Time 2z 8z 4z 5z 




200 














100 















00 












100 X 












S 


20(fc 












SfiJ 


300 \ 












400 \ 













Fig. 75. — History of Pressure Head h, starting from 
Half Opening. 





L 


H 


m 


m 1 


z 


T 


Case a 


4000 


2700 


•01 


•02 


20 


z 


„ b 


>> 


,, 


•01 


•02 




52 



Thus with the data of Fig. 48a and-with"[m =-5 the value of v = 
7-594, while v 1 =7*731, leaving only a residual v 1 — v =-137 to be 
realized by the further opening of the valve. In this case the 
values of h are negligible, only reaching a maximum of 2-4 feet. 



144 



HYDRAULICS OF PIPE LINES 



10QQ. 




\2z >4z JBz Ite l.Oz 1.2z 1.4z 1.6z 
Time 
Fig. 76. — History of Pressure Head h for Various Points 
in Pipe Line (Closure). 

L H m m 1 z T 
4000 200 -10 00 20 z 

Case a at valve 

„ b „ 1000 feet from 
„ c „ 2000 
„ d„3000 „ 
„ e„3600 M 



>> »> 




6z 



Wz 



2z 3z 
Time 

Fig. 77. — History of Pressure Head h for Various Points in 
Pipe Line (Closure). 

L H ra m x z T 

4000 200 -10 00 2 lOz 

Case a at valve. Origin at £ = 

„ b „ 1000 feet from „ „ 0„*=-25sec. 

„ c„ 2000 „ „ „ „ 0„ <--50 „ 

„ d„ 3000 „ „ „ „ 0„ *=-75 „ 

For better comparison the four cases are brought to a common origin of 
time. Actually these origins are displaced relatively by quarter seconds, as 
noted above. 



WATER RAM OR SHOCK IN WATER CONDUITS 145 

On the other hand, with the same data and with m=-10 and -05, 
the initial velocities are 5-216 and 3-13, and the values of h at full 
opening are 39-2 and 64-0 respectively (see Fig. 71). 

In cases where the ultimate valve opening is small compared 
with the size of pipe, the initial velocity will vary nearly with the 
initial opening, and in such cases the ultimate drop in pressure at 




Wz 



~Ez 7z ~8z 
Time 

Fig. 78. — History of Pressure Head h for Various Points 
in Pipe Line (Closure). 

L H m m x z T 

20,000 3000 -02 -00 10 lOz 

Case a at valve. Origin at t = 

„ b „ 5000 feet from „ „ „ * = l-25 

„ c „ 10,000 „ „ „ „ „ * = 2-50 

„ d„ 15,000 „ „ „ „ „ *=*3-75 

For better comparison the four cases at e brought to a common origin of 
time. Actually these origins are displaced relatively by time intervals, as 
noted above. 




Fig. 79. — History of Pressure Head h for Various Points 
in Pipe Line (Opening). 

L H m m x z T 
4000 500 -00 -05 2-0 z 

Case a at valve 

,, b „ 1000 feet from „ 
„ c „ 2000 „ „ „ 
„ d „ 3000 „ „ „ 

H.P.L.— L 



146 



HYDKAULICS OF PIPE LINES 



full opening within the time t=z, will vary approximately with 
the amount of valve movement or nearly with the velocity (v 1 — v ) 
to be acquired (see Fig. 72). 

With cases intermediate between these extremes, the results 
will be of the same general character, but varying in less direct 



9z 10z 




Fig. 80. — History of Pressure Head h for Various Points 
in Pipe Line (Opening). 

L H m m x z T 
4000 500 -00 -05 2-0 lOz 

Case a at valve. Origin at t = 

„ b „ 1000 feet from „ „ 0„*=-25 

„ c „ 2000 „ „ „ „ „ i=-50 

„ d „ 3000 „ „ „ „ „ t=-16 



1000 



Scale of m 




Fig. 81. 



Az Time .8z 1.2z 1.6z 2z 

-History op Pressure Head h with Assumed Partial 
Reflection at Valve (Closure). 

L H m m 1 z T 
4000 500 -05 -00 2-0 2z 
Case a Percentage of reflection at valve -00 



•60 
1-00 



The dotted line shows the course of the curve with full reflection for full 
closure in time z. 



Water ram or shock in water conduits hi 



1000 



Scale of m 



.05 
800 


04 


.03 


.02 


.01 


600 










400 








a 


200 






^~7T 















00 



.4z Time .82 1.2s 1.6z 2z 

Fig. 82. — History of Pressure Head h with Assumed Partial 
Reflection at Valve (Closure). 

L H m m x z T 
4000 500 -05 -00 2-0 2z 

Case a Percentage of reflection at valve (m — m)/m 
» b „ „ „ „ „ s/v 



300 



Scale of m 




Fig. 83. — History of Pressure Head h with Assumed Partial 
Reflection at Valve (Closure). 

L H m a m, z T 



Case a 
„ b 
,, c 
„ d 



4000 500 -05 -00 2-0 5z 
Percentage of reflection at valve 



•00 
•25 

•50 

•75 

100 



148 



HYDKAULICS OF PIPE LINES 



ratio with valve movement as the final opening is large in com- 
parison with the area of pipe. In all cases, however, the maximum 
drop will vary roughly with the velocity (v x — v ) to be acquired. 

Subsequent to the arrest of valve movement at full opening, 
the head h will return to zero either by periodic lifts or through 



300 



Scale of m 




Time 2z 3z 4z 5z 

Fig. 84. — History of Pressure Head h with Assumed Partial 
Reelection at Valve (Closure). 

L H m m 1 z T 
4000 500 -05 -00 2-0 5z 

Case a Percentage of reflection at valve (m — m)/m Q 
„ b „ ,, ,, „ „ s/Vq 



Scale of m 
.01 .02 




4z 5z 

Fig. 85. — History of Pressure Head h with Assumed Partial 
Reflection at Valve (Opening). 

L H m m 1 z T 
4000 500 -00 -05 2 5z 

Percentage of reflection at valve 



Case a 
„ b 

„ c 

,, e 



•00 
•25 
•50 
•75 
1-00 



WATER RAM OR SHOCK IN WATER CONDUITS 149 

alternating plus and minus values, as the characteristics may 
determine, and in accordance with the same general relations as 
outlined for the case of opening from full closure. 

Maximum value of h. In this case the maximum value of h 
(drop in pressure) is found at the end of the valve movement or 
when t=T. It may therefore be computed from equation (75) 
by substitution of the proper values. 

Valve Opening from Initial Partial Opening. Time T Greater 
than Time z for Double Traverse of Acoustic Wave.— The effect of 
extending the time in these cases is similar to that in opening from 
full closure, as discussed above. fSuch increase of time reduces 

Scale of m 
.01 .02 




z Time 2z 3z 4z 5z 

Fig. 86. — History of Pressure Head h with Assumed Partial 
Reflection at Valve (Opening). 

L H m m x z T 

4000 500 -00 05 2-0 5z 

Case a Percentage of reflection at valve (m 1 —m)/m 1 
„ b „ „ „ „ „ {v x — v)/v t 



the ultimate or extreme drop in pressure and modifies the general 
character of the curve beyond the point where t=z, as indicated 
in Figs. 73, 74, 75. 

After the close of the valve movement, the value of h will return 
to zero in the same general manner as for other cases and as previ- 
ously noted. 

Maximum value of h. In this case the maximum value of h 
(drop in pressure) is sometimes found when t=z, or again when 
t=T, or in some cases between t=z and t=T. It will therefore 
result from an application of (75), as may be required. 

Pressure History at any Point in the Line. — Following the 
method outlined in Sec. 38, Figs. 76-80 show the history of h at 
various points in the line for selected cases with hydraulic character- 
istics as noted. 

Partial Reflection at Valve. — Due to the aksence of any defi- 
nitely assured basis for estimating the proportion of reflection at 



150 HYDKAULICS OF PIPE LINES 

the valve end of the line, no attempt has been made to work out 
any considerable number of numerical cases. In Figs. 81-86 are 
shown a few cases in illustration of the application of the method 
to a numerical problem. Figs. 81-84 show the form of the curves 
in the case of valve closure, and Figs. 85, 86 correspondingly for 
valve opening. In Figs. 83, 85, for the interest attaching to extreme 
conditions, curves have been worked out on the assumption of 
a constant value of /=0 and/=-25, though doubtless in most cases 
the actual value is greater than the larger of these. The value 
/=0 implies reflection from the upper end of the line, but no reflec- 
tion from the valve end. These various curves will repay careful 
study in connection with those for /=l-00 or full reflection at the 
valve, and which are given on the same diagrams for convenience 
of comparison. 

43. Approximate Formulae 

Due to the inherent complexity of the relations involved in the 
general problem of shock or water hammer, there have been many 
attempts to find approximate or working formulae which might be 
sufficiently accurate for most practical cases. There is perhaps no 
problem connected with practical hydraulics for the treatment of 
which a greater number of approximate or working formulse have 
been propqsed. The difficulty of developing any such formula 
which shall be fairly general in its application is, however, shown 
by the extreme divergence among the results given by the applica- 
tion of these various formulae to the same problem. It is true that 
by introducing special limitations or conditions a formula may be 
developed which shall be relatively simple, and at the same time 
fairly accurate for the special conditions implied. All such formulae 
lack generality, and it is just here that danger enters in connection 
with their use. The author of a proposed method or formula may 
well understand its limitations and the conditions under which it 
may safely be used ; but once the formula has found a place in 
engineering literature, these limitations are apt to be forgotten 
and such formulae are often proposed or used for cases to which 
they are entirely inapplicable. Under such circumstances the 
results obtained are wholly misleading, and their use implies a 
confidence which is quite without foundation in actual fact. 

Due to these considerations it has seemed desirable to note, in 
the present section, some of the better -known formulae which have 
been proposed and which are in current use in varying degrees. 

In this discussion there is implied no criticism of these formulae 
as such or of their authors, but rather an attempt to show the 
relation between such formulae and the more complete theory 
developed in the preceding sections, and thus to indicate the general 
conditions under which such formulae may be safely or appro- 
priately employed. 



WATEK RAM OR SHOCK IN WATER CONDUITS 151 

The Allievi Formula. — The general equations developed by 
Allievi* rest upon the same fundamental assumptions regarding 
the physical phenomena involved as the equations and methods 
of Sees. 36 and 38. Allievi's development is, however, based on 
certain restrictions or limitations as noted below. 

Applied to the same problem and with the same interpretations 
and conditions, the equations of Sees. 38, 41 give identically the 
same results as those of Allievi. A somewhat different method of 
development has, however, been here preferred by reason of the 
better picture which it enables us to form, of the physical state of 
the column of water within the pipe and along which an acoustic 
wave is travelling back and forth with reflection at the ends, all 
as developed in detail in preceding sections. 

The restrictive conditions which are assumed in the development 
of the Allievi equations are as follows : 

1. The omission of the influence due to friction. 

2. The omission of the head v 2 /2g corresponding to the main 

pipe line velocity v. 

The latter is usually small compared with the other quantities 
involved, and its omission is a matter of no serious importance. 

The former is of serious or minor importance according to circum- 
stances. For closure the influence of the friction head is relatively 
large at the start and decreases as the pipe line velocity v decreases, 
vanishing with v=0 at complete closure. The omission of the in- 
fluence due to friction will affect the course of the history of h 
during the closure and the final value to some extent, but in many 
cases not seriously. 

In general the error due to the omission of friction will be more 
and more important as the friction head is a larger and larger 
fraction of the total head H. 

In the case of opening, the influence of friction is minimum at 
the start, when, as a rule, the greatest drop in pressure occurs, and 
increases, reaching a maximum with the final steady motion velocity. 
If the opening is from complete closure, the influence of friction 
on the maximum drop is relatively small ; if from part opening, 
the error due to its omission will be more significant, increasing in 
importance as the friction head is a larger and larger fraction of 
the total head H. 

These general equations of Allievi have, however, for the most 
part, been put aside in favour of an approximate formula pro- 
posed by him and used currently by many engineers without a 
proper understanding of its derivation or limitations. These are 
in brief as follows : 

We may write equation (42) showing the composition of h in 
terms of the successive values of s as follows : 

's— Sj+Sa— s 3 +etc.\ 



h=a(* 



(80) 

Annali della Societa degli Ingegneri," Rome, Vol. XVII, 1902. 



152 HYDRAULICS OF PIPE LINES 

Put F=s— 5 1 +5 2 — Sg+etc. 

Then remembering the special notation of Sec. 38 we shall have 

F 1 =s 1 — s 2 +s 3 — etc. 
Hence (80) becomes 

h^F-Fj. 
It is also evident by inspection that 

S=F+F V 

Now it is evident that if F increases uniformly with the time so 
must also F v and so must also their sum or s. Likewise it is evident 
that F—F 1 then becomes a constant quantity, or &=constant. 

It was then assumed that a uniform rate of decrease of valve 
opening, that is a uniform rate of decrease in m, as used in the 
present system of notation, will during the period from t—z to t—T 
result in a uniform rate of increase in F, and hence in a uniform 
rate of decrease in v and in a constant value of h. 

While it was recognized that these conditions will not be realized 
with mathematical precision, it was assumed nevertheless that, 
for practical purposes, the time histories of v and h during the 
time period from t=z to t=T, may, with uniform rate of valve 
closure, be taken as showing a substantially uniform rate of decrease 
for v and a substantially constant value for h. 

Similarly for opening, it was assumed that with a uniform rate 
of valve area increase, the history of h after the first drop, will 
show a partial return followed by a nearly uniform value for the 
remainder of the time period up to t—T. 

A formula is then developed for the determination of this assumed 
uniform value of h during the time period from t=z to t—T. 

The formula itself is developed in terms of the ratio between 
the total pressure head, which we may here denote by y, and the 
original head H. In terms of the present notation we have then 



y—H^-hl omitting 



2g) 



Next let x=Yi 
Jtl 



Then the following equation in x is deduced 
a 2 -z(2+rc 2 )-fl=0 
and of this equation we have the two roots 



a=l + 2^±-v / ^ 2 +4) ( 81 ) 

where n=^j 

Of the two values in (81) the + sign applies to the case of closure 
and the — sign to that of opening. 

Regarding the two roots given by the + and — signs of the 
radical, it may be noted that their product is 1, so that if either 
is known the other may be immediately found. 



WATER RAM OR SHOCK IN WATER CONDUITS 153 

The limitations in the proper use of this formula arise from the 
fundamental assumption that a uniform rate of valve movement 
will, for this period, determine a uniform rate of decrease in v or 
a uniform value of h. In some cases this will hold, at least approxi- 
mately. In others, as for example, in Fig. 49 the rate of decrease 
of v is widely divergent from uniformity, and correspondingly the 
history of h shows no approach to a flat top or nearly uniform 
value during the time period t=z to t=T. Generally speaking, if 
the friction head is a large part of the total head, the history of h 
shows no approach to the form assumed for the Allievi formula, 
and the application of the formula to such a case will give a result 
widely divergent from that indicated by the more complete theory. 

In the case of valve opening, furthermore, the formula does not 
profess to give the value of the first extreme drop, but rather the 
mean of the subsequent history between t=z and t—T. Here 
again, however, where the final value of m is large, the history of 
y will show small similarity to that assumed in the formula, and 
in such case likewise the use of the formula will lead to a result 
having but remote relation to that indicated by the more complete 
theory. 

Warren's Formula.* — Complete Closure. If we assume that the 
conditions of closure are such that the excess pressure rises from 
according to a linear law during the time z=2L/S and then remains 
constant, we have the same general conditions as assumed in the 
Allievi formula. Warren, however, does not assume necessarily a 
linear rate of valve area closure, and rests these general assumptions 
regarding pressure change rather on a certain amount of observa- 
tion of actual, cases with valve areas controlled by governing 
devices in normal power plant practice, and in which cases the 
change in pressure closely fulfilled a program such as assumed. 

With this assumption the principle of impulse and momentum 
may be invoked to obtain a simple approximate value of the 
assumed constant excess head reached during the -time period from 
t=z to t=T. 

Considering the mass of water in the pipe and neglecting the 
relatively small amount which flows out during closure we have as 
follows : 

The average force acting over the cross section of pipe during 
time from £=0 to t=z is Awh/2. 

The uniform force acting during time from t=z to t=T is Awh. 

The total change of momentum produced is LAwvJg. 

Then remembering that the sum of the products of force by time 
during which it is in operation will equal the change in momentum 
produced, we have : 

Awhz A *. m LAwv 

— - — \-Awh(T— z)— - 

* 9 

* "Trans. Am. Soc. C.E., 1915," p. 238. 



154 HYDRAULICS OF PIPE LINES 

Whence j/f-?\JQ* 



or h= ,*"* x (82) 



T -i) 



g\ t- 



This is Warren's formula. 

Obviously the limitations in the use of this formula are sub- 
stantially the same as for the Allievi formula. Both assume (how- 
ever realized) substantially the same form of time history for h and 
where the circumstances are such as to realize approximately such 
a form of time history, the formula should apply with close approxi- 
mation. 

In other cases the results will be in error more and more widely 
as the form of history for h departs more and more widely from that 
assumed. 

If again, in this formula, we should assume T very long relative 
to z then the formula will reduce approximately to 

fc=gf° (83) 

which has been proposed as a formula for water hammer conditions. 
i^This is equivalent to assuming the value of h uniform throughout 
the entire period of closure, and on this assumption the formula may 
be directly deduced on the principle of impulse and momentum. 

It will be noted that this value is one -half that given by the 
Joukovsky formula. Obviously both cannot be correct, at least for 
the same case. As noted under the next heading the Joukovsky 
formula applies to one assumed history of h and that of (83) to an 
entirely different assumed history, while that assumed by Warren 
and Allievi is a combination of the two. 

Any one of these histories might by chance be approximated to 
in actual experience, but a study of the diagrams of h as developed 
by the use of the more complete theory indicates how small the 
probability of any one case conforming to any one of these assumed 
histories, and in consequence the remote chance that any one of the 
formulae based thereon could be safely employed in any given case. 

Joukovsky' s Formula.* — It is assumed (however realized) that 
the rate of decrease of velocity is uniform, or in other words that 
the rate of increase of s is uniform. In such a case, as we have seen 
in Sec. 38, the history of h will show a series of slopes up and down 
from a minimum of to a maximum determined by the value 
reached at t=z. We have then simply to find the value at this 
instant ^=zas the maximum value reached during thejnovement. 
i§ The rate of increase of s will be vJT per unit time. 

The rate of increase of h will be, then, av /T per unit time. 

* "Memoirs Imperial Academy of Sciences, St, Petersburg, 1897," 
Vol, IX. 



WATER RAM OR SHOCK IN WATER CONDUITS 155 

This rate is uniform from £=0 to t—z when the maximum value 
is reached. Hence for such maximum value we shall have 

, zav n 2Lv, 



o 



(84) 



T gT 

which is Joukovsky's formula. 

Vensano* has extended this to a pipe line of varying diameter in 
the form 2(L 1 v 1 +L 2 v 2 +L 3 v 3 -{-etc.) 

h ~ JT~ (85) 

Where L v L 2 , etc., denote the lengths of the various sections and 
v l9 v 2 , etc., denote each the initial velocity of flow in these various 
sections. 

The limitations on the use of this formula spring from the special 
assumption made regarding the form of the time history of v. 
In Allievi's formula the assumption is made that the change of 
velocity is uniform during the period t=z to t=T. In Warren's 
formula the assumption relates to the form of the time history of h, 
but agrees in form for the time period z to T, with that assumed by 
Allievi. In the present case the assumption is made of uniform 
rate of decrease of v from the start of the movement. It is only this 
assumption which can justify the form of time history for h which 
furnishes the basis for the formula, and it is of special importance 
that such conditions should be fulfilled for the time period to z. 

It may be shown, however, that at the most the condition 
assumed can only be approximately realized, and in no case will a 
uniform rate of valve closure determine a uniform rate of velocity 
decrease, especially in the early part of the movement and hence in 
the time period to z. 

At the best, therefore, the special condition assumed as a basis 
for the development of this formula can be only imperfectly 
realized and in many cases the departure will be extreme. 

Generally speaking the greater H relative to L and the smaller 
the area of the valve opening compared with pipe cross section, the 
more nearly will the conditions here contemplated be realized. In 
inverse cases the departure from the conditions assumed may 
become so wide as to vitiate the formula for practical use. 

Formulae Based on Principle of Mass and Acceleration.— Several 
writers f have proposed formula for change of pressure in pipe lines, 
based on the principle of mass and acceleration and considering the 
body of water within the pipe as forming a single mass subject to an 
accelerating head measured by h. 

In the development of such formulae we may conveniently start 
with the fundamental equations (33), (36) as follows : 

v=mu (86) 

Mu 2 =H+h (87) 

* "Trans. Am. Soc. C.E., 1918," p. 185. 

f See particularly A. H. Gibson, " Water Hammer in Hydraulic Pipe 
Lines." New York : D. Van Nostrand. 



156 HYDKAULICS OF PIPE LINES 

and to which instead of (34) we add the equation : 

%J^ (see Sec. 11 (n)) (88) 

at L 

Assuming also uniform valve area closure, we have 

m=(m —kt) (89) 

The combination of these equations should serve to give a com- 
plete solution of the problem, but, unfortunately, the resulting 
equation does not seem to admit of integration and reduction in 
analytical terms. 

If, however, the influence due to friction is omitted, the equations 
become amenable to treatment. In such case M becomes a constant 
and equal to 1 /2gf and for (87) we have 

u*=2gf(H+h) (90) 

If then we combine (86), (88), (89), (90), eliminating v and h, we 
shall have the equation : 

nil 

u 2 -2fkLu-2gfH=-2fL(m -kt)^ (91) 

For 2gfH put its value u 2 . 

Also put fkL=P 

and ^P*+u 2 =B. 

We may then reduce (91) to the form 

du dt 



(92) 



(B+P-u) (R-P+u) 2fL (m -kt) 
Reducing the left-hand member into partial fractions and 
integrating, we find 

. R+P— u] u R, m —kt 

Evaluating the above equation between u and u and reducing, 
we have 

' &+P-u \ / R+P-u \ R,_/m { 

R-P+u) 8 [R-P+uJ^P 



- J l^)..(93) 



R+P-u _ R+P-u / m -kt \f 
R-P+u~R-P+u \ m J K } 

From (93) or (94) it follows that when £=0, u=u . Again at the 
close of the full movement when t=m lk we shall have 



u=R+P= ^/p2+u 2 +P (95) 

From the form of (94) it is seen that u steadily increases during 
the movement of the valve, from its value u =2gfH at £=0 to its 
maximum (R-\-P) just at the instant of complete closure. 

The value of h follows directly from (90) : 

h =ir H (96) 



WATEE RAM OR SHOCK IN WATER CONDUITS 157 

and hence h will increase continuously with u and will reach its 
maximum value with u at the instant of complete closure, or in 
the case of partial closure at the final instant of valve movement. 

In the case of complete closure k=m /T and with u=(R-\-P) we 
readily reduce (96) by substitution of values, to the form 

h^(R+P) (97) 

For the case of opening, the analytical treatment is in effect 
contained within that for closure as above. It is only necessary to 
introduce a change of sign in dvjdt and in k, and to suitably interpret 
the terms of the preceding equations. 

We shall have then for opening, 



R—P—u_R—P—u / m \p 
R+P+u~R+P+u \m +kt) 



(99) 



and h=H-^j (100) 

If the valve starts from a nearly closed position then m is nearly 
0, and toward the close of the valve movement m l(m -\-kt) will 
become nearly 0. Hence in equation (99) the last term approaches 
as a limit as t increases, while correspondingly u will approach the 
limit (R—P), as shown by the other members of the equation. This 
value is readily seen to be less than u . From the form of (99) it 
thus follows that u steadily decreases during the movement of the 
valve and with small m approaches the value (R—P) as a limit 
when t is large. Hence, as shown by (100), there will be a con- 
tinuous increase in the value of h (drop in head), and the maximum 
value will be reached at the close of the time T when the movement 
under consideration is completed. 

From (100) we derive in the same manner as for (96) the maximum 
value of h, corresponding to u=(R—P) in the form 

hJ^{fi-P) (101) 

For two cases, one of closure and one of opening, with the same 
values of P and R, and denoting opening and closure by sub- 
scripts 1 and 2, we have for the limit values of u and h : 

Ul u 2== E 2 -P 2 =u 2 =2gfH (102) 

We also readily derive the relation : 



)^f(hlf (103) 



(JcLU(\ _J 



It must be remembered, however, that these limit values of h 
imply closure either complete or nearly so, and opening from full 
closure or nearly so. 



158 HYDRAULICS OF PIPE LINES 

The rapidity of approach of u and h to these limit values will 
depend on the circumstances of the case. Generally u and h will 
approach more quickly and more nearly to their limit values with 

Head H or u Q . large 
T . . . long 
Valve area . small relative to pipe cross section. 

In the case of valve opening, as shown by (98), with small values 
of m , the value of h rapidly and closely approaches its limit value, 
and as m becomes smaller and smaller the rapidity of approach 
becomes rapidly greater. When ra =0, that is when the valve 
starts from the shut position, the approach to the limit value is 
instantaneous, at least so far as indicated by these equations. In 
such case, therefore, the value of h would hold substantially uniform 
during the period of valve movement at the limit value, and the 
velocity in the pipe line would be given substantially by the 
equation : v=mu=kt{R-P) (104) 

It should be noted that this value applies only in the case of the 
valve starting from the closed position. 

As the value of m is relatively larger ; that is, as the valve 
starts from a position more and more open, the growth of h (drop in 
pressure head) is relatively less abrupt and the actual value realized 
at time T is a smaller fraction of the limit value, as in (101). 

A comparison of (97), (101) with the Allievi formula, equation 
(81) leads to the surprising result that when expressed in terms of 
the same quantities the two formula are identical. Equations (97), 
(101) express the maximum or limit values of h based on principles 
of mass and acceleration alone and without reference to the forma- 
tion or propagation of acoustic waves. Allievi 's equation starts 
with the same fundamental theory as developed in Sees. 38, 41, but 
as the result of the process of development with the introduction of 
special or limiting assumptions, the same ultimate expression as in 
equations (97), (101) is reached. 

Obviously the same limitations regarding the use of (97), (101) 
apply, as in the case of the Allievi formula, and as noted in connec- 
tion with that formula. 



CHAPTER IV 

STRESSES IN PIPE LINES 

44. Introduction 

In approaching this subject we must first inquire as to the sources 
of load capable of producing stress. 
These are as follows : 

1. Internal pressure. 

(a) Balanced. 
(/3) Unbalanced. 

2. Weight of pipe line or element under consideration. 

3. Weight of water contained in pipe line or element under 

consideration. 

Balanced internal pressure implies uniform pressure in opposite 
directions over equal projected areas, as in the case of a completely 
closed chamber or on opposite sides of a pipe. It is well known 
that such a load, in an element with circular cross section, pro- 
duces tension alone. Where the cross section is non- circular, bending 
moments involving tension, compression and shear will be com- 
bined with the tension arising directly. 

Unbalanced internal pressure implies a condition of unbalanced 
force so far as the element under consideration is concerned, such 
unbalanced force tending to displace or separate the element from 
the remainder of the system or from its environment or attach- 
ments. The stress developed by such unbalanced pressure may 
be tension, bending or cross breaking, shear, or compression and 
in all combinations according to the details of the case. 

In discussing stresses developing in these various ways in pipe 
lines we find it convenient to consider different combinations of 
conditions as follows : 

Two conditions regarding distribution of pressure. 

(a) Static or no flow. 

(b) Steady flow. 

We may properly assume that in all cases the transverse dimen- 
sions (diameter usually) will be so small compared with the head 
involved as to permit of assuming the pressure uniform over any 
given cross section of the pipe or pipe line element. Regarding 
vertical extension, however, we may have the two cases : 

(c) Element or system under consideration substantially in one 

horizontal plane. 

159 



160 HYDRAULICS OF PIPE LINES 

(d) Element or system under consideration extending over 
considerable differences of horizontal level. 

It results in case (ac) that the pressure may be taken as uniform 
over any section involved and in general throughout the system. 

In case (be) the differences in pressure at different sections will 
be due to changes of area and hence of velocity, and not to changes 
of level. Also we may take the pressure as uniform over any given 
section. 

In case (ad), depending on differences in level, there will be 
differences in static pressure throughout the system. 

In case (bd), likewise, there will be differences in pressure due 
to changes in level as well as changes in velocity. 



45. Ring Tension in a Cylindrical Pipe or Element 
with Circular Cross Section 

We assume static conditions (a) above. The pressure under 
conditions of flow will be less than under conditions of rest. In 
this case, furthermore, the question of vertical extension enters 
only as a factor in the intensity of the pressure. 
Let D— diameter (i). 
t— thickness (i). 
p=internal pressure (pi2). 

T= actual working stress in longitudinal joint (pi2). 
c= efficiency of longitudinal joint. 
Then the familiar formulae of mechanics give us the following : 

pD) 



■(I)" 



2et 
2teT 

pB 
f ~2e2\ 

It should be noted that whenever a pipe or pipe line is under 
pressure, the ring tension stress is always operative. 



46. Longitudinal Stress 

We assume conditions (a) as above, and for the reasons cited in 
connection with ring tension. The full longitudinal stress in a 
pipe is only developed when there is a cap or closure across the 
section or in the case of bends, angles or turns ; and in all cases the 
pipe must be sufficiently free from external constraint to permit 
longitudinal movement under the end pressure developed. A 
typical case of longitudinal tension is furnished by a pipe closed 
at the end with the end free to move. 



STRESSES IN PIPE LINES 161 

Under these conditions the formulae of mechanics give us 



T- 


pD 


(pi2) 


P= 


= 7T 


(pi2) 


t= 


pD 


(i\ 


leT 


v 1 ; 



(2) 



The notation in these equations is the same as for ring tension, 
except that T and e must be taken as relating to the circumferential 
joint. 

These latter equations show the well-known fact that, other 
things equal, the longitudinal stress on a circumferential section 
is just one -half the circumferential stress on a longitudinal section, 
or that the ring tension is just twice the longitudinal stress. 



47. Stresses Due to Angles, Bends and Fittings 

p In approaching this subject it is first necessary to^consider the 
character and measure of the load which may be thrown on a 
pipe line as a result of the internal pressure on such elements of 
the line. The load will be represented by certain unbalanced forces 
acting between the angles, bends or fittings and the remainder of 
the line, and as a result of which, stress will be developed either 
in the line itself or in some form of anchor or tie which is intended 
to carry such load and thus relieve the line. It will be necessary 
here to consider the four combinations of conditions, (ac), (ad), 
(be), (bd), as noted at the opening of the chapter. 

Case (ac). Static Conditions with Influence Due to Differences o! 
Level Insignificant. — It is well known from mechanics that a com- 
plete enclosure under uniform internal fluid pressure is in complete 
equilibrium under the forces developed by such pressure. That is, 
no condition of internal fluid pressure can develop any force tending 
to move the enclosure as a whole. Parts or elements of pipe fines, 
however, are not completely closed. Such a part or element of 
a pipe line system presents therefore an incomplete enclosure under 
internal fluid pressure, and as a corollary from the equilibrium of 
a complete enclosure it follows that an incomplete enclosure will 
not necessarily be in equilibrium. It follows, further, that the 
forces required to maintain the part or element in equilibrium will 
be represented exactly, both in magnitude and direction, by the 
fluid pressures over the areas which would be required to com- 
pletely close such part or element. This concJusion applies to any 
part or element of a pipe line such as an individual pipe or part 
thereof, an angle, Y branch or valve body. Hence we deduce the 
following broad proposition. 

Assume any part of any pipe line system under uniform pressure. 

H.P.L. — M 



162 HYDKAULICS OF PIPE LINES 

Then the resultant forces, due to internal fluid pressure, will be 
represented in magnitude and reversed direction by the fluid pres- 
sure over the areas which would be required to make of such ele- 
ment or part a complete enclosure. Conversely the forces or 
system of forces required to maintain such an element or unit in 
equilibrium will be represented in magnitude and direction by the 
fluid pressures over the areas required to produce complete closure 
and without reversal. 

This may be illustrated as in Fig. 87. Let AC denote a straight 
uniform length of pipe. Then the areas required to close the 
system would be represented by AB and CD. Hence the force 
required to maintain equilibrium would be represented by the 
fluid pressures on the sections AB and CD. But these are equal 
and opposite. Hence the force is zero and the section of pipe is in 
equilibrium, a well-known fact which common sense readily tells us. 



A C 


^ -N,-fc 


] B D \Ss 




Fig. 87. — Resultant Force on Pipe Line Elements — Static 
Conditions. 

Again, take the part between|(7 and E. Here the closing areas 
will be sections CD and EF. The pressures are equal and oppo- 
site in direction, but do not act along the same line. Hence to 
maintain equilibrium there will be required a couple with GH as 
arm. The resultant force on the section itself is, of course, measured 
by a couple equal in amount and opposite in direction. The re- 
sultant force on the section is therefore a couple measured by 
P7tD 2 /4:xGH, and tending to turn the element in a clockwise 
direction. 

If instead of a pipe of uniform section we have a difference 
between the two areas as at AB and CD (Fig. 88), we shall have 
two forces opposite in direction, but unequal in value, and the 
resultant will be a force and a couple. 

Again, take the element of line between EF and I J (Fig. 87). 
Denote for convenience the total pressure over a cross section of 
the pipe by P. Then the balancing system will be represented by 
two forces P and P applied at the centres of the sections EF and 



STKESSES IN PIPE LINES 



163 



IJ. These may, of course, be readily combined into a single re- 
sultant measured by 2P sin 0/2. This reversed will then be a 
measure of the resultant acting on the element and along the line 
KL. 

Again, with an element containing a Y branch such as IJMO, 
we shall have three sections IJ, MN, OQ and three forces, and 




Fig. 88. 



-Resultant Force on Pipe Line Elements- 
Static Conditions. 



a combined resultant according to the methods of elementary 
mechanics. 

Again, suppose we should consider a part of a straight length 
contained between two oblique imaginary planes such as EF, 
OH (Fig. 88). Then the closing areas will be two ellipses of area 
(nD 2 cosec 0)/4, and the forces required for equilibrium will be 




v\\U//// 



Fig. 



V 

-Distribution of Resultant Load 
along Pipe Line Bend. 



measured by the pressure p acting over this area. This will give 
the forces IJ, KL, or by transfer and composition, a single resultant 
MN. The force exerted on such an element will then be equal 
and opposite, or MP. 

Distribution of Unbalanced Pressure along a Pipe Bend.— The 
analysis developed above serves to determine the unbalanced load on 
the bend as a whole. It becomes of interest, however, to determine 



164 HYDKAULICS OF PIPE LINES 

likewise the law of distribution of this unbalanced load \ along 
the line of the bend. Thus referring to Fig. 89, let AB denote the 
bend with short straight lengths at each end. Let CDEF be a 
small element of the bend and 8 the corresponding angle. Then 
from the principles above developed we shall have for the resultant 
unbalanced pressure on this element a force SF measured by 

8Q=2pA sin A? 

where p= pressure in pipe and A— cross sectional area. Now if 
8 is small, sin 8 0/2 approaches 8 6/2, and hence for a small element 
weshaUhave SQ=pASd. 

Now let Ss=length of arc of mid curvature for the element and 
p the radius of curvature. Then 8 0= 8s j p. Hence we have 

se=^ ( 3) 

P 
At the limit when 8 and 8s become differentials this relation 
becomes exact, and we have 

dQ=^ (4) 

P 

Expressed in words this tells us that the bend under these con- 
ditions is subject to a distributed load applied in the direction of 
the radius of curvature and proportional at any point directly to 
the product pA and inversely to the radius of curvature p. In the 
case of a circular bend p will be uniform, the radius of the bend. If 
non-circular, the intensity of the load will vary inversely as the 
values of p. Furthermore, it appears that for any small element of 
length along the mid- curvature line M, the resultant load is 
measured by the product pA multiplied by the length of the ele- 
ment, and divided by the radius of curvature p. 

Thus in the case of a bend 24 inches mean radius in a pipe 10 
inches internal diameter, and under a pressure of 100 (pi2) the un- 
balanced pressure load per inch of length on the mean radius will be 

5_ 100x78-54x1 , 

8Q= ^4 ==327 (P)- 

A pipe bend under these conditions is therefore to be treated 
simply as a curved beam subject to a distributed load determined 
in accordance with (3) or (4), and with end reactions determined 
as previously developed for the bend as a whole. The case becomes 
reduced, therefore, to the mechanics of the beam and need not be 
considered in further detail here. It may be pointed out, however, 
that in the case of a bend of uniform radius attached rigidly to the 
tangent pieces at the ends of the bend, the unit load, measured as 
above, will result in longitudinal tension only, measured by the 
total load pA . That is, while the condition of the bend is exactly that 
due to a distributed load applied along the radius and measured 



STRESSES IN PIPE LINES 



165 




Fig. 90. — Resultant Force 
on Pipe Line Elements — 
Static Conditions. 



as in (3) or (4), nevertheless the result of this is, when the ends are 

anchored, to produce simply a tensile stress in the bend. It is in 

fact readily seen that the case is entirely similar to that of a chamber 

of circular cross section under fluid pressure ,**and in which,^as is" well 

known, the shell is subject to tension 

only. In the case of the bend the 

load pA/p takes the place of the 

pressure in a cylindrical shell, and 

with the result of a pure tensional 

stress measured by the load pA divided 

by the cross section of the metal. 

If the ends of the bend are not 
rigiolly attached to the rest of the line, 
or in any event if they are not con- 
strained by forces having longitudinal 
components and which can therefore 
balance the end pulls pA, the case 
becomes entirely changed and cross - 

breaking stresses may develop. Further reference will be made to 
this case in Sees. 49, 50. 

Case (ad). Static Conditions with Influence Due to Differences of 
Level Significant. — Let AG (Fig. 90) denote a part of a pipe line in 
which we cannot assume the pressure as uniform throughout. 
Assume ideal planes AB and CD forming a complete enclosure 
of the part under consideration. Now from hydrostatics we know 
as follows : 

1. In a completely closed chamber under hydrostatic pressure 

there is no resultant force in a horizontal direction. 

2. The only resultant force is in a vertical direction, and it is 

measured by the weight of the liquid. 

3. The centre of application of such a resultant gravity force is 

at the centre of gravity of the liquid or at the centre of 

volume of the enclosure. 
It follows that if the chamber of enclosure is not completely 
closed, the total unbalanced system of forces will consist of two 
parts : 

1. The gravity component vertical in direction, equal to the 

weight and passing through the centre of volume of the 
chamber. 

2. Pressure components represented by the pressure over the 

various openings, reversed in direction. 
Thus in Fig. 90 the resultant will consist of the vertical gravity 
component G acting downward, the pressure component P x over 
the area AB acting to the right and the pressure component P 2 over 
the area CD acting upward. The unit pressures over AB and CD 
will in this case be different, and P x and P 2 must be computed 
accordingly. We are therefore led to the following general propo- 
sition, which indeed will include both cases (ac) and (ad). 



166 



HYDRAULICS OF PIPE LINES 



Given any part of a system of pipes, connections, fittings, etc., 
under hydrostatic pressure. Then for such part of the system we 
may represent the unbalanced force system as follows : 

(1) Draw a vertical line downward through the centre of volume 

under consideration, and of length proportional to the 
weight of the water or fluid contained. 

(2) Assume imaginary planes closing all openings, thus giving, 

constructively, a completely closed volume. 

(3) Through the centre of pressure of each such area draw a line 

at right angles to the plane of the opening directed from 
without inward, and of length proportional to the total 
pressure on such area. In ordinary cases the centre of 




Fig. 91. — Resultant Force on Pipe Line Elements- 
General Case with Flow. 



pressure may, without sensible error, be taken at the centre 
of gravity of the area. 

The system of unbalanced forces will then be represented by the 
forces denoted by the lines (1), (3) above, which may be combined 
into a single resultant or a single resultant and a couple or treated 
as may be desired by the methods of elementary mechanics. 

Case (be). Steady Flow Conditions with Influence Due to Differ- 
ences of Level Insignificant. — In order to determine the reaction 
between moving water in a pipe and the containing pipe, we 
may conveniently resort to a fundamental principle of hydraulics 
which may be stated as follows (see Appendix III). 

In any hydraulic system containing water in motion, and where 
the dimensions are such that we may neglect the weight of the 



STRESSES IN PIPE LINES 



167 



water as such, the force reaction of the water on the system will be 
given by the vector sum of the following systems of forces : 

(1) The total pressures over the ideal sections bounding the 

system or element, reckoned from without inward and 
combined as vectors. 

(2) The sum of the momenta per second at inflow and outflow, 

the former taken direct and the latter reversed and all 
combined as vectors. 
In the case of water at rest system (2) disappears and we have 
left simply system (1) which reduces the case to one of static equi- 
librium as discussed under case (ac), and by the aid of precisely the 
same principle. 

This proposition holds for cases including friction, so that if we 
know the conditions of flow, the force reaction between the water 
and any part or the whole of the system may readily be determined. 




Fig. 92. — Resultant Force on Pipe .Line Elements- 
Flow from Open Reservoir. 



As an illustration of the proposition in this general form let 
ABDC (Fig. 91a) denote an element in a pipe line system through 
which water is flowing in the direction from AB to CD. Let p, v, A 
and M with subscripts 1 and 2 denote respectively the pressure, 
velocity, area and momentum at the sections AB and CD. 

From the centre of the section AB we then draw lines EF and EG 
respectively in the line of flow and perpendicular to the plane of the 
section. Lay off on these lines distances EF and EG, representing 
respectively M 1 =wA 1 v 1 2 /g and Pi^PxA^ and find the resultant 
EQ. A similar construction at CD gives the resultant HS. The 
two resultants intersect at 0. Then lay off OQ 1 =EQ and OS 1 —HS 
and find the final resultant OR as representing in magnitude and 
direction the total reaction of the water on the element in question. 

Usually the bounding sections AB and CD may be taken per- 
pendicular to the lines of flow, simplifying the construction as in 
Fig. 916. 

Certain special applications will be of further interest. 



168 



HYDKAULICS OF PIPE LINES 



Thus in Fig. 92 let the pipe AC be discharging through the area 
CD— A with velocity v. In this case in order to complete the stream 
line system we must imagine stream lines continued from A and B 
up to the surface of the water as indicated by dotted lines. Then 
assuming discharge at CD into the atmosphere and measuring 
pressure above the atmosphere, we shall have p=0 over both faces 
NN and CD. 

Again the momentum at NN will be sensibly zero while that at CD 
will be M=wAv 2 /g. The resultant of systems (1) and (2) will there- 
fore be a force wAv 2 /g directed from right to left, or in the direction 
CA, and measuring the reaction of the water on the system. 

These relations are indicated by the vector diagram where 



r N N 






^ ■■ ,„ — ■-- i^-^_ .!_ _=£~=: ; 


A 




~7l 

\ » 


F 






B 


G 



R 



a -o 





Q 



Fig. 93. — Resultant Force on Pipe Line Elements- 
Flow from Open Reservoir. 



Oi£=reaction of water on system and 0©=opposite and equal 
reaction of system on water. 

Furthermore, if secondary losses are neglected we shall have 
v 2 /g=2H and F=2wAH= twice the static pressure over the area 
of discharge. 

This is an expression of the well-known relation that neglecting 
losses due to friction and turbulence, the dynamic pressure due to a 
jet or stream is twice that of the static head necessary to produce 
the velocity of discharge. 

If we have an elbow in the discharge line, as at C (Fig. 93) then 
using the same method as before we find the total force reaction 
measured by OR—F=wAv 2 jg and as indicated in the vector 
diagram (a). 

For all cases where the system considered extends from a reser- 
voir where the water is at rest, to one or more points of free discharge, 
it is clear that system (1) of the forces enumerated above will dis- 
appear and that the force reaction on such system will be measured 



STKESSES IN PIPE LINES 



169 



by force system (2), the resultant of the forces representing the 
reversed momentum per second at the point or points of discharge. 
Thus let Fig. 94 represent a plan view of a system including reservoir 
and three discharge outlets. Draw lines from outlets 1 and 2 in the 
line of discharge and extending back to a point of intersection 0. 
Then lay off forces OA and OB to represent the reversed momentum 
per second represented by the discharge from these orifices. Find 
the resultant OP and draw the line through the discharge orifice 3 back 
to an intersection X . Then from X lay off O x P x —OP and X G— 
reversed momentum per second for orifice 3. Then the resultant 
X R will represent the final resultant force reaction on the system. 
These methods may be applied to any combination of elements 
involving a reservoir and one or more discharge orifices. 

3, 




>--2 



Fig. 94. — Resultant Force on Pipe Line Elements- 
Case with Multi-Discharge. 



Let us consider next the elbow G (Fig. 93). If we assume free 
discharge into the atmosphere the pressure over the section FG will 
be sensibly zero. Hence system (1) of the forces will disappear and 
we shall have the vector diagram at b, where OA— vector value of 
M at FG } Oi?=reversed vector value at DE, and OR— force 
reaction on elbow. Note again that OR will here be twice the value 
of the displacing force found for the same elbow under static 
conditions. 

Next let the elbow G be fitted with a nozzle DE (Fig. 95). This 
will reduce the velocity of flow through the pipe and elbow and 
decrease the friction loss with corresponding increase in pressure. 

The total force reaction will be found in the same manner as for 
Fig. 93, but its value will not be the same. The relation between 
the force reaction and the size of the orifice will be considered at a 
later point. 

Taking first the elbow G between FG and IJ we shall have over 



170 



HYDRAULICS OF PIPE LINES 



the two areas FG and IJ a pressure P=pA i sensibly the same in 
numerical value and related as at OA, OB in the vector diagram at a. 
We also have M at inflow direct represented by AG and M at out- 
flow reversed represented by BD. These combine together and give 
a final resultant OR for the force reaction on the elbow. 

To find the value of OC=P-\-M we have, in the general case, 
including friction, the relation 

w~ r 2g~ r C 2 r ' 



Then P= P A=«>a(h-£^ 

But M=wA-. 
9 

Hence OC=P+M=wA[H+v*f~ 



L 

'0 2 i 



;> 



(5) 




A M C 

Fig. 95. — Resultant Force on Pipe Line Elements — 
Discharge erom Open Reservoir through Elbow 
and Nozzle. 



This equation shows that the value of P-\-M will increase or 
decrease or remain independent of v according as the parenthesis 
(Il2g-L/G 2 r)is +, -, or 0. 

Again, it should be noted that the total resultant OR may be 
viewed as the resultant of the two components (P-\-M) } one hori- 
zontal and the other vertical, acting jointly on the elbow G. 

Next if we include the elbow and nozzle as one element, we shall 
have the same horizontal components P and M denoted by OA and 
AC (Fig. 956). 

For the pressure at DE we shall have zero and for the reversed 



STRESSES IN PIPE LINES 171 

momentum a value M 1 represented at OB, greater than M by 
reason of the smaller area and higher velocity. We shall then have 
the final resultant OR representing the total reaction on the combina- 
tion of elbow and nozzle. 

Again, if we lay up OD (Fig. 956) —OB of a we shall have the 
difference DB as the measure of the reaction between the nozzle and 
the elbow. 
To determine the measure of these forces we proceed as follows : 
Let a — area of nozzle. 
a/A=m. 

/= efficiency of nozzle. 
u= velocity through nozzle. 
v— velocity along pipe. 
We have M^wau 2 jg—wAmu 2 jg '. 

We may find u as in Sec. 17 and thus express M ± in terms of the 
conditions of flow. This will give 

wAH (2mf) 

M x = , , 2gfLm* (6) 

i_t " C*r 

We may next express (P-\-M) as in (5) but substituting for v its 
value mu, determined as in Sec. 17. This will give ■ 



P+M=wAH 



m 

1 + - 



»(■-£) " 

2gjXm*_ 



(7) 



Denoting the denominator in (6) by B and reducing we may 
express these values as follows : 

„ _ wAH(2mf) 
Ml ~ B 
' P | M= waH(l+m*f) 
B 

Hence (P+M)—M 1 = v ' J J - (8) 

The values of m and / are always less than 1, and it is readily 
shown in such case that the parenthesis in (8) is always positive. 
Hence P-\-M is always greater than M x and the difference gives the 
reaction of the nozzle, which is always downward as we should 
expect. 

An interesting question arises as to the condition which will make 
M x a maximum in any given case. It will be clear that with a very 
large or m nearly 1, the velocity through a will be limited by the 
friction loss in the line. With a small value of a the velocity along 
the line will be less, the friction loss less and the discharge velocity 
greater. It may thus result that the maximum value of M v the 



™=y^ 



172 HYDRAULICS OF PIPE LINES 

reaction on a line fitted with a discharge nozzle, will not be found at 
the point of full opening. 

Taking M 1 as in (6) and considering m a variable we readily find 
by the usual method for maximum and minimum the value 

2gfL' 

This gives the value of a in terms of A for the maximum value of 
M v It is also readily shown when this condition is realized that 

u^VW^ (10) 

2 *=V1 < n > 

and ™- =friction head=^ (12) 

It is clear that, as defined, m is always less than 1. Hence in 
order to make such a maximum value of the reaction possible, we 
must have, from equation (9) L>G 2 r/2gf. Otherwise as m is 
increased from we should simply have a continuously increasing 
value of the reaction up to the point oim—l or full opening. 

Again, since the impulse of a stream on a fixed bucket, as in the 
case of an impulse water-wheel when the latter is at rest, is directly 
proportional to the reaction on the nozzle and pipe line, it follows 
that the above values determine the conditions for maximum 
starting torque on an impulse water-wheel. It is of interest to 
compare these relations with those of Sec. 20 giving the conditions 
for maximum power. 

Case (bd). Steady Plow Conditions with Influence Due to Differ- 
ences of Level Significant. — This is the same as the last case, but 
with the addition of the effect due to gravity on the water over 
considerable variations in level. 

Reference is made to this case in Appendix II. We proceed as in 
the case (6c) but add a fourth system of forces representing the 
weight due to gravity acting vertically downward with the other 
systems as in the preceding case. 



48. Load Due to Weight of Pipe or Element, 
and also of contained water 

The weight of the pipe or element and also that of the contained 
water must, of course, always be supported. As to whether they 
will join in forming a stress-producing load in such degree as to 
require recognition will depend on the circumstances and geo- 
metrical arrangement of the pipe system and its supports. Generally 
speaking, the effect of gravity in producing pressure within the pipe 
line or pipe line element will be accounted for by the principles 
and methods already developed. When, however, the horizontal 



STRESSES IN PIPE LINES 



173 



dimensions of the element or system under consideration are of any 
considerable extent, the weight of the element as a structure and the 
weight of the contained water must be included with other forces 
due to unbalanced pressure in order to find the final load. These 
loads due to weight are readily found and combined with the load 
due to unbalanced pressures by the principles of elementary 
mechanics. 

49. Stresses in Expansion Joints 

The principles previously developed find an important application 
in the case of expansion joints. Thus if the joint is made up, as in 
Eig. 96, with the internal diameter for the flow of the water equal 




Fig. 96. — Expansion Joint. 




Fig. 97. — Expansion Joint. 



throughout to that of the pipe^except for the part|between AB and 
CD, then it is clear that there will be a force tending to separate the 
two members of the joint and measured by the unit pressure p 
multiplied into the area of the annular ring of metal at CD. In the 
case of large pipes this might rise to a force of very considerable 
magnitude. Thus if diameter =48 inches, thickness of metal = 
1 inch and p=100(pi2), we find the total load as above =15,000 
pounds. , r , ;* \\"C 

If the metal of partf-F is relieved between A and E as indicated 
by the dotted lines, such relief will make no difference in the 
resultant load tending to separate the parts of the joint. It is 
readily seen that the pressures on the two additional shoulders thus 
formed will be equal and opposite and will balance each other on 
the piece F. 



174 HYDRAULICS OF PIPE LINES 

On the other hand, if the outside diameter of the sliding member 
H is made equal to the internal diameter of the pipe, as in Fig. 97, 
it is clear that there will be no axial force on the part F due 
to the liquid pressures at the joint, while the axial pressures on the 
part H will just balance and there will therefore be no resultant 
force tending to separate the two parts of the joint. 

If, again, the internal diameter of the pipe D is made greater 
than the outside diameter of the slide H, the liquid pressures will 
give a resultant tending to push H into F t producing a tension which 
must be carried by the pipe lying to the right. 

These cases all develop as simple applications of the principles 
already discussed, and no matter what may be the form or design 
of an expansion joint, the use of these principles will serve to give 
the resultant of the liquid pressures at the joint. 



50. Combinations of Bends or Elbows with 
Expansion Joints 

Cases of some interest may develop as the result of the combina- 
tion of expansion joints and bends or elbows. In considering such 
cases it must be borne in mind that the expansion joint virtually 
cuts the pipe at the joint and that no longitudinal stress or support 
or constraint can be transmitted from one member of the joint to 
the other — at least so long as no additional guard bolts or other 
members are fitted. Actually, expansion joints are very commonly 
supplied with bolts connecting the two members and allowing a 
certain limited degree of freedom, but preventing the complete 
separation of the two parts. In this manner, and between fixed 
limits, changes due to temperature may be accommodated while the 
two members of the joint cannot become entirely disconnected. 

Holding in mind the inability of the expansion joint by itself to 
transmit longitudinal force, together with the hydraulic principles 
discussed in the preceding sections, the various cases will admit of 
simple treatment. 

Thus in the case of a bend, as at AB (Fig. 98a) , fitted with expansion 
joints at A and B, the resultant force on the bend will be readily 
found by a simple application of the principles of Sec. 47, case (ac) 
or (be), as may be required. This force will then be represented by 
some resultant KL and which must be carried by reactions at the 
ends A and B. These end forces or reactions cannot be longitudinal. 
They will instead develop as side pressures between the two parts of 
the expansion joint. Taking these reactions at right angles to the 
pipe at these points, we have then the bend acting as a beam under 
a hydraulic unbalanced pressure load distributed over the curved 
portion, and supported by two forces acting at right angles to the 
pipe at the expansion joints. The total amount of the unbalanced 
pressure load is readily found as developed in Sec. 47, and this will 



STRESSES IN PIPE LINES 



175 



serve, as in the usual manner with beam problems, to determine the 
end reactions at A and B. The distribution of the load is deter- 
mined as in Sec. 47 and is represented by a force acting along the 
radius equivalent to a pressure pA/r per unit length of arc, where p 
is the pressure in the pipe, A the cross-sectional area and r the 
radius of the bend. 

In this manner the distribution of the load and the end reactions 
become known. The entire problem becomes, therefore, reduced to 
the mechanics of the beam, and need not be here treated in further 
detail. 

It may be stated, however, without present proof that the 
maximum bending moment at the middle point of a circular bend 




Fig. 98. — Expansion Joints Combined with Elbow. 



as developed by the application of the mechanics of the beam to 
this problem is shown to be 

, /l— COS0 

iff: 



. /I— COS0\ 

=prA I — ) 

V cos J 



where r=mean radius of bend and 0=half angle of bend. 

For a 90° bend or elbow, as in Fig. 98a, this becomes : 
M=-4tl42prA. 

Thus with a 90° bend of 24 inches mean radius in a 10 -inch pipe 
under a pressure of 100(pi2) and with the ends carried in slip joints, 
we shall have for the maximum bending moment at the middle" of 
the bend the value : 

M =-4142 x 100 x 24 x 78-54=78,000. 

With metal about J -inch thick this would result in a stress of 
about 1800(pi2) in the outer fibre, a stress, therefore, by no means 
serious in itself. 

In order that an elbow, as in Fig. 98a, may have support as 
assumed, the pipe must be tied or supported near the slip joints ; 
otherwise the parts of the joint would separate completely. 

If an elbow is fitted with a single slip joint, as in Fig. 986, the 
condition as regards bending moment on the elbow is indeterminate. 
This results from the fact that the end reaction at A is indeterminate, 
depending on the stretch and flexibility of the pipe at B. The total 



176 HYDRAULICS OF PIPE LINES 

force on the elbow will, however, be represented by a resultant KL, 
found as in Sec. 47, and the component of this parallel to DA will 
give a force which will tend to separate the parts of the joint and 
which must be carried by some tie or support near B or else by guard 
bolts at A preventing movement beyond a certain limit. 



51. Case of a Long Pipe with Open Ends Carried 
in Slip Joints 

An interesting case is presented by a slightly bent pipe, such as 
AB (Fig. 99), with the ends carried in slip joints and not maintained 
rigidly in line. We may thus assume the possibility of the pipe's 




Fig. 99. — Resultant Force on Long Elastic Pipe. 

assuming the form of an elastic curve as indicated in the figure. The 
balancing forces in this case will be represented by two forces P, P 
acting over the end sections and directed from within outward. 
Hence the resultant force on the pipe itself will be represented by 
two equal forces PP acting over the end sections and directed 
opposite, or from without inward. This places the pipe exactly in 
the condition of a curved column carrying on the end a load P. 
We may also consider the pipe as in the condition of a beam 




Fig. 100. — Resultant Force on Long Elastic Pipe. 

loaded transversely with a distributed load, the resultant of which 
will be the same as for the two forces PP applied at the ends as 
above noted. 

The law of distribution of this load has been determined in 
Sec. 47 and is given in equation (4). We have, therefore, for the 
transverse load Q per unit length along the pipe or beam the value 

0=- (13) 

P 

or, as noted, the transverse load varies along the line of the bend or 

curve, inversely as the radius of curvature of the bend. 

Turning again to the pipe as a whole, the resultant of the two 



STRESSES IN PIPE LINES 



177 



forces P acting on the ends and along the line of the axis at the ends 
will be given by a construction, as in Fig. 100, and measured by 

B=2Psinld. 

This resultant will tend to bend the pipe still further, and the 
actual result will depend on the relation between the magnitude of 
the resultant E, the law of the distribution of the actual forces along 
the length of the pipe, and the resistance of the latter to bending. 
The pipe may therefore be considered as in the condition of a 
column subjected to an end thrust P and liable to yield by buckling ; 
or otherwise, to a beam subjected to a transverse load varying 
inversely as the radius of curvature of the pipe. 

In order to develop the question of the relation of the strength of 
the pipe to the magnitude of this transverse load, we may employ 
the well-known relations in the theory of elastic beams between load, 
shear, bending moment, slope and deflection. 

These are embodied in the following equations : 

Let Q=load per unit length. 
S= shear. 

M = bending moment. 
m=tangent of slope from straight line. 
y= deflection from straight line. 
i"=moment of inertia of section. 
i£= coefficient of elasticity. 
x = distance along pipe. 
.L=length of pipe. 
p= unit pressure in pipe. 
D= diameter of pipe. 
£=thickness of wall." 



Then S=fedx 

M=^Sdx 



™=m\ Mdx 



y=\mdx 



(14) 



Or conversely, 
dy_ 



dx 

d 2 y_dm_ M 
dx 2 ~~dx~EI 
d*y_d 2 m_ 1 dM_S 
dx*~ dx 2 ~EI dx~EI 
d*y_d*m_ 1 d 2 m_ 1 ds_Q_ 
dx*~ dx* ~~EI dx 2 ~~EI dx~EI 



(15) 



H.P.L. — N 



178 HYDRAULICS OF PIPE LINES 

Now when the radius of curvature p of a beam is large and the 
curvature small, as we may properly here assume, we have, by a 
well-known property of plane curves, ™ 

For a curve such as Fig. 101a, where y 
is taken positive, downward, d 2 yjdx 2 and 
hence p will be negative. The negative 
sign here is to be considered simply as 
having a directive significance. Hence 
in expressing Q in terms of P and p 
from (13) we must give p a negative sign 
in order that Q and P may both be con- 
sidered positive. Having this algebraic 
relation in mind, we have, therefore, 




Fig. 101. 



_f=_p^ (16) 



Resultant Force on Long P ^x 

Elastic Pipe. Also we have in general (see (15)) : 

« =JW § < 17) 

It will be noted that of these two equations (16) is restricted to 
the special case which we are here considering, while (17) is general. 
Combining we have 

#y _P_d*y 
dx*~ EI dx 2 
Whence integrating twice, we have 

S-- & ™ 

The solution of this equation is the well-known sine curve, which 
we may take in the form 

y—a sin — (19) 

Li 

Then differentiating twice, we have 

d 2 y «7T 2 . 7TX 

- 2--£ <»> 

Comparing (18) and (20), we have 

P_7t2 

EI~~L 2 
or P=p4=_ 



STRESSES IN PIPE LINES 179 

Let D x and D 2 denote the external and internal diameters of the 
pipe and t the thickness of metal. Then we shall have 

V=D?L* < 21) 

«(iV-iy ) 

64 

Where t is very small compared with the diameter we may take a 
mean value between D x and D 2 and reduce the value of / to /= 
ttDH/S. Substituting in (21) and reducing we find the approximate 
value : 

4- 935ffP* . 
P= L 2 < 22 ) 

Turning now to (19) as the equation to the form assumed by the 
pipe, and taking successive derivatives, we have from (15) the 
following : 

y=a sm -=- 

7T 7TX 

m=a y cos ~t~ 
Ju h 

__ _,.. 7T 2 .KltX 

M=—EIa,Y- 9 sin T 

S=—EIa 1 r-o cos - T - 

"- 4 7ra; 



Q=Elajx sin -^ 



These are illustrated in Fig. 101 a, 6, c, d, e, and show that the 
successive values of load, shear, bending moment, tangent of slope 
and deflection are given by alternate sine and cosine curves with 
coefficients as indicated. 

If now we assume the relation of (21) to obtain we may put for 
EI its value in terms of P and thus find for M , S, and z the following : 

M =—Pa sin -?=:—Py 
Li 

S——Pa-= cos — 
h Li 

It will be seen that the mid-length value of y is a. It is also seen 
that p in (22) is independent of a. That is, the value of P or p in 
order that the assumed conditions may subsist is independent of 
the mid-length deflection a. 

This means that no matter what the deflection, so long as it is not 
enough to involve any marked curvature of the pipe, that is, so 



180 HYDRAULICS OF PIPE LINES 

long as we may consider the value of p to be given by d 2 y/dx 2 , so 
long will the pipe remain in neutral equilibrium in the form of a 
sine curve under the constant value of P or p as given above. 

That is if the pipe is given any mid-length deflection a, so long as 
a is relatively small, the pipe may be expected to assume a sine 
curve with a for the maximum deflection and to remain in equi- 
librium under the forces operating. If then a is increased or 
decreased, so long as it still remains relatively small, the pipe will 
remain wherever it is placed and in equilibrium under the forces 
operating. 

This value of p may therefore be taken as a critical value, defining 
the condition for neutral equilibrium under the general conditions 
assumed. If p is less than the critical value, then the pipe, if 
slightly displaced from a straight line, will return by the operation 
of its elastic forces. If p is greater than the critical value, then the 
deflection will increase beyond limit and flexure will result, at least 
unless other conditions step in to prevent. This limit value of the 
pressure is relatively high. Thus let X>=3 inches, £=120 inches, 
£=•1 inch. Then from (22) we find p=2879(pi2). 

The indications of the formulae of the present section, and in 
particular of (21), giving the value of the critical load, have been 
verified experimentally by Fidler* with both copper and drawn 
steel pipes respectively 1-17 inches and 1 inch outside diameter by 
1-00 inch and -90 inch inside diameter, and 10 feet long. 



52. Influence of Anchors, Piers, Ties, Abutments, 
etc., on the development of stress in plpe 
Lines 

In the preceding sections we have examined the various sources 
of load which may enter into the production of stress in pipe lines 
and pipe line elements. It is clear, of course, that these loads "must 
all be carried in one way or another, but it does not follow that they 
will all enter fully into the production of stress on the pipe line 
itself. This will depend very largely upon the manner in which the 
pipe line is supported or constrained, and hence upon the extent to 
which such loads may be carried in whole or in part by the piers or 
anchors or other means of support or constraint. 

Loads arising from balanced internal pressure must, in general, 
be carried by the pipe, or pipe line element subjected to such 
pressure. Loads arising from unbalanced internal pressure are 
very commonly carried, in some part at least, by various external 
means of support or constraint. If such unbalanced force is 
localized at a known point, such as the reaction from a stream 
issuing from an opening, the support or constraint can be applied in 

* " Calculations in Hydraulic Engineering." Longmans, Green and Co., 
London and New York. 



STRESSES IN PIPE LINES 



181 



> 



the line of the resultant of such force, thus relieving the pipe line 
itself of any cross -breaking load. If the forces producing load of this 
character are distributed over a considerable extent of the line, as 
for example the weight of a horizontal line of pipe with its con- 
tained water, then support will naturally be supplied at appropriate 
intervals, thus placing the line in the condition of a continuous 
girder with stresses developed accordingly. 

It is clear that parts of the system separated by points of com- 
plete constraint will represent independent systems so far as these 
various forces and the resulting stresses are concerned. Hence in 
determining the stress due to loads resulting from unbalanced 
forces, the following general program may be followed : 

1. Note the separation of the system as a whole into parts by 
points of complete constraint. 

2. Taking any one part thus set off, determine in magnitude, 
direction and line of application, the various 
unbalanced forces due to internal pressure. 

3. Determine also the forces due to 
gravity on the pipe and its contents, in 
case such are of importance for the problem 
in hand. 

4. In case there is but one point of con- 
straint capable of resisting the forces in any 
one plane, then the problem with reference 
to such forces is entirely definite. The 
loads are known and they must all be 
carried at the one point of constraint. The 
problem from this point on is therefore one 
of the mechanics of materials and need not 
here be further considered. 

5. In case there are two or more points 
of constraint which might share in resisting 
the forces in any one plane, then the 
problem is entirely indefinite as to the part 
of the load carried by the pipe and that Fig. 102. — Reaction on 
carried by the points of constraint, and Support. 
hence indefinite as to the stress developed 

at any point in the pipe due to the loads. In order to reach any 
definite result some assumption must be made regarding the manner 
in which the constraint is shared among these various points. With 
such an assumption made, the problem becomes one of simple 
mechanics as before. 

A few simple illustrations will serve to show the application of 
these general principles. 

In Fig. 102 suppose CD vertical and EF horizontal with AB as a 
point of support and constraint. Then the weight will be carried 
at AB as a direct load. Again the reaction due to the escaping jet 
at F will be represented by a force acting along the line FEG. This 




182 



HYDRAULICS OF PIPE LINES 



will place the pipe in the condition of a cantilever beam with load at 
the end as determined by the magnitude of the reaction due to the 
jet. This reduces the problem to one of simple mechanics. 

Again, in Fig. 102 suppose that CDEF is horizontal. Then there 
will be a vertical load perpendicular to the plane of the paper due to 
gravity acting on the pipe and its contents and a horizontal load 
acting along the line FG. This will give a resultant load in an 
oblique direction with the pipe acting as a cantilever beam. 

Again, if we assume CD horizontal but the nozzle turned vertically 
down, the reaction of the latter will oppose gravity and we shall 
have the net resultant in a vertical line with the pipe acting still as 



B 



^ 



^ 



Fig. 103. — Reaction on Support. 



a cantilever. If in the last case the nozzle is turned up instead of 
down, the two loads will be additive instead of subtractive in their 
relation, with the pipe acting as a cantilever. 

In these various cases, of course, the line of action of the resultant 
of the gravity forces will pass through the centre of gravity of the 
pipe and its contents, and that of the reactive forces along the line 
of flow at the nozzle backward from the outlet. 

Precisely the same methods apply in the case indicated in Fig. 103. 

In Fig. 104 the same arrangement is indicated as in Fig. 102 with 
the addition of an abutment of some sort at G. This will place the 
pipe in the condition of a beam fixed at O, supported at G and 
loaded at D. The force at C is readily determined by taking 
moments about G while the bearing reaction at G will be the sum of 
the forces at D and C. Precisely the same relations will obtain in 
case of a tie at G instead of a strut or abutment. 

In case the point G is located at D, the point C will be relieved of 
load except as some stretch of the tie or yield of the abutment may 



- STKESSES IN PIPE LINES 



183 



permit a cross -breaking load on the pipe. The condition, therefore, is 
indefinite except as some assumption is made regarding the extent 
of stretc x h or yield, or regarding the degree in which the load is thus 
divided between the external constraint and the pipe. 

In a case such as that of Fig. 105, with two points of support, 
one on either side of the elbow, the condition is definite if the 
supporting forces are in the nature of flexible ties. In such case the 



Ai 



n 



B 



CM 




D 



E 



Fig. 104. 
Reaction on Support. 




Fig. 105. 

Reaction of Elbow on 

Flexible Ties. 




Lcb 




Fig. 106. 

Reaction of Elbow on 

Direct Support. 



Fig. 107. 

Reaction of Elbow on 

Constraining Piers. 



force D alone can oppose the load OA and the force C alone can 
oppose OB. This condition will therefore definitely determine 
known total tension loads at C and D. 

If, however, the elbow is supported against an abutment F 
(Fig. 106), the case becomes indefinite. If F can be assumed to carry 
the entire load, there will be no load transmitted to the pipe to be 
carried at any other point. Otherwise, as F may yield in some 



184 HYDRAULICS OF PIPE LINES 

degree, load will be thrown on the pipe which must be carried at 
some other point. 

Again, if the supports at G and D (Fig. 107), are of the nature of 
anchor blocks giving complete constraint, then the problem becomes 
indefinite. Either point is capable of carrying the load due to the 
elbow, and hence the actual amount carried at each point will be 
indefinite except as some special assumption is made. 

Such cases of partial support or constraint are often furnished in 
the case of buried pipes by the weight of earth cover or by the 
resistance to lateral or longitudinal movement furnished by earth 
filling. 

In all such cases of uncertain distribution of support or con- 
straint, judgment alone can be relied on to determine some reason- 
able basis of division with a safe margin to allow for the measure of 
uncertainty involved. 

In the case of cast-iron pipe with bell and spigot joint, especial 
attention must be given to the proper support or constraint of 
elbows in order to prevent the danger of separation at the joint. This 
is of particular importance in high-pressure fire lines at hydrant 
settings and elsewhere where the forces developed might seriously 
endanger the integrity of the line. To safeguard such points 
suitable strap and rod ties are commonly fitted, preventing move- 
ment of such a character as to render possible the opening up^of the 
joint. 

53. Stresses in Connections and Fittings 

In the preceding paragraphs we have considered the subject of 
pipe line elements, connections and fittings with special reference to 
the unbalanced forces which may develop, and with emphasis on the 
loads which such forces may throw on the pipe line itself. We have 
now to consider briefly the stresses which may develop in the 
connections and fittings themselves. 

From the hydraulic standpoint there is no sharp line of demarca- 
tion between the pipe and a fitting or connection, such as an elbow, 
angle, Y branch, Tee, valve or nozzle. They are all parts of a 
continuous water conduit and hence subject to the same funda- 
mental hydraulic laws and principles. We therefore proceed to the 
determination of the stress on any section of such an element by 
considering the part lying between such section and the nearest 
point of constraint, and determining the load on such part by the 
principles discussed in the preceding paragraph. If the part thus 
cut off by the section under consideration is without constraint, the 
problem is simplified by the elimination of any considerations of the 
share of the load which the constraint may carry. From the load 
thus determined, the stress inyfche section is then to be^determined by 
the usual methods of the mechanics of materials. 

In many cases the parts may be considered as without constraint 



STRESSES IN PIPE LINES 185 

in one or both directions at right angles to the sections under con- 
sideration. In such cases the tension over any such section due to 
the direct loading resulting from the internal pressure will result 
as follows : 

Let ^4=projected area of surface subjected to pressure on the 
side of the section not subject to constraint. 
a = area of section (metal). 
2?=unit pressure in chamber. 

Other notation as above. 
Then from the principles already adduced we have 

pA=Ta. 

or T=^ (23) 

a 

Thus if a constant value of T is to be maintained we must main- 
tain a constant relation between A and a for the various sections 
which we may cut across the chamber. This will usually be neither 
practicable nor desirable on account of the existence of other 
stresses, as we shall see below. In any event it will result that the 
maximum value of T will be found where there is the maximum 
value of Ala', or otherwise, on the section where the area of metal 
is least in proportion to the projected area subject to load. 

In any such case, therefore, it is only necessary to seek out by 
trial the section where the area of metal is least in proportion to 
the projected area subject to load and to find T as above. This 
will give the maximum direct stress in tension due to internal 
pressure. 

In all such chambers of irregular form, such as Y branches, valve 
bodies and elbows, there will be certain sections which are non- 
circular in form. In Y branches in particular, certain parts approach 
a flat or only slightly curved form. In all such cases the stress 
along any filament lying between two such parallel sections will not 
be wholly tension. A bending stress with its accompanying shear 
will be set up, and the unit stresses from these must be combined 
with the stress in tension due to direct loading. Usually the forms 
of such chambers are not such as to permit of direct investigation 
and indirect and approximate methods must be used. We have 
two reference forms to which the actual forms may usually be 
approximated. These are the ellipse as a form of section and the 
flat plate. 

When parallel sections for some little distance are nearly uniform 
in size and form and approximate to an ellipse, we may apply the 
formulae for the strength of a pipe of elliptical section.* 

These are as follows : 

Let a and b denote the two semi-axes of the ellipse (see Fig. 108). 

* "Resal Traite de Mecanique," Vol. V, p. 134. 



186 HYDRAULICS OF PIPE LINES 

Assume a dimension of unity or 1 inch in a direction perpendicular 
to the plane of the section. 

Let c =excentricity=-\/l — (b/a) 2 . 

C=a constant depending on the ratio of b to a and given 
approximately by 

<7=.333-f-167~ 
a 

M A = bending moment at end of long axis. 

i/ B =bending moment at end of short axis. 

p=unit internal pressure. 

Then we have 

M A ^G (24) 

M B =^(1-C) (25) 

In the above equations the inch and the pound are the units. 

The bending moment at A will develop tension on the inside 
and compression on the outside, while that at B will develop ten- 
sion on the outside and compression on the inside. At A, therefore, 







V 



D 

Fig. 108. — Stress on Elliptical 
/ l\ C Section. 

the tension due to the direct load (p over the diameter AC) must 
be added to the tension on the inner fibres due to M A , in order to 
give the total maximum tension at A, which will be on the inner 
fibres. Similarly at B the tension due to the direct load (p over the 
diameter BD) must be added to the tension on the outer fibres 
due to M B> in order to give the total maximum tension at B> which 
will be on the outer fibres. 

By the use of these equations an approximate value may be 
developed for the maximum stress in any such elliptical section. 

Where the form approaches that of a flat plate over a certain 
area, the necessary support is usually developed by the use of a 
system of intersecting ribs, thus forming a series of cells, usually 
triangular or four-sided, the flat base of which is intended in each 
case to be self-supporting as a flat plate. In this mode of design, 
therefore, the ribs are intended to act as girders in carrying the 
load, while the elementary flat or nearly flat areas thus formed 
are expected to be self-supporting between the ribs, and to con- 
tribute to the support of the load as a whole. 



STRESSES IN PIPE LINES 187 

Two questions thus arise : 

1. The thickness of the flat plate between the ribs. 

2. The dimensions and spacing of the ribs. 

In many cases, however, the thickness is fixed by other con- 
siderations or by relation to other parts, and question (1) only 
remains. 

The design of these features is necessarily by empirical formulae, 
as follows : 
Thickness o! Flat Plates supported by Ribs. 
Let ^4=areaof cell or element supported by ribs at boundary, 
(i2). It is here assumed that no two dimensions of 
A differ widely. 
p= pressure (pi2). 
t= thickness (i). 

C= constant, about 100 for cast iron and 160 for cast steel. 
Then 

t=^f- (26) 

Supporting Ribs. — Regarding the design of the supporting ribs, 
two questions arise : 

1. The spacing of the ribs. 

2. The dimensions of the ribs. 

Only the most general indications can be given regarding these 
matters, and reference should be made, if possible, to successful 
designs of similar character and operating under generally similar 
conditions. 

The spacing of the ribs is usually made from 8 to 16 or 18 inches, 
varying with the size. The thickness should be not far from the 
thickness of the body or shell and the height may vary from 4 to 6 
or 8 times the thickness at the highest part, often tapering or fading 
off to nothing at the ends, according to the features of the design. 
As a general guide, the results to be applied with judgment, use may 
be made of the formula : 

V&LJ1 (27) 

Where p= pressure (pi2). 

^4= area of a strip considered as supported by one rib 
through from one end to the other. Normally A 
will equal the product of the distance between the 
ribs by the extreme length (i2). 
L= length of rib from end to end (i). 
k= stress developed in outer fibre (pi2). 
/= moment of inertia of section of rib (i4). 
y = distance from neutral axis to outer fibre (i). 
In taking values of k, I and y , the thickness of the metal con- 
stituting the body or shell may be added to the height of the rib 
proper. Thus for illustration, suppose an area 15 inches wide and 



188 HYDRAULICS OF PIPE LINES 

60 inches long considered as supported by a rib 2 inches thick and 
10 inches total height at centre. What will be the safe pressure, 
allowing a working stress of 16,000 pounds in the steel ? We find 
pAL/24:=2250p and kl/y = 533333. Whence p=231 (pi2). 

As noted above, however, the results of no formula alone should 
be accepted without judgment and comparison with similar cases 
if possible. 

54. Stresses of Joint Fastenings, Flanges, 
Bolts, etc. 

Stresses in joint fastenings may be either tension or shear. 
Wherever a longitudinal pull develops in a pipe line, the joint 
fastenings must, of course, carry such pull as a load in tension or 
shear, depending on their disposition relative to the joint — shear 
in a riveted joint and tension in a flange joint with bolt fastenings. 
Similarly where a bending stress develops, the fastenings will be 
thrown into either shear or tension. In all such cases the principles 
of the present chapter will serve to determine the load at the joint, 
at least so far as it is determinate, and the problem is thereby reduced 
to one of the mechanics of materials and may be treated accordingly. 



55. Stress due to Bending Moment in Spans 

A long pipe line supported at frequent points, insofar as its 
relation to deformation through gravity forces is concerned, operates 
as a continuous girder. 

The mechanics of a continuous girder or beam shows that the 
maximum bending moment occurs at the pier (considered as a 
point of support), and is 

M^ (28) 

where W = weight between supports and L=length. We have, then, 
from the familiar beam formula : 

S-s « 

Where k = stress in outer fibre of beam. 

/^moment of inertia of section. 
y = distance from neutral axis to outer fibre. 
For a cylindrical shell about a diameter, we have 
T _ itDH 
8 
D 

Also W= — 1- o-nDtL (pounds). 



STRESSES IN PIPE LINES 189 

Where all dimensions are in inches, w— weight of one cubic inch 
of water and cr== weight of one cubic inch of steel. 
Hence we find 

k-nDH _ WTtD*L* airDtL 2 

4 "~ 48 H 12 - 

or fc= _+^- 



'\12r3Df 



But w=-0361 and cr=-2835. 

Substituting these values we have 

*=(^+^( P i2) (30) 

In the preceding formulae the values of / and of w are expressed 
on the assumption that the thickness of the pipe is small compared 
with the diameter. This assumption is usually permissible in pipe 
line problems. It may be noted, however, that the value of D used 
should correspond to the mid-thickness of the shell. If higher 
accuracy is desired or if t is not small compared with D then we 
must use the following : 

64 

4 4 

where D x and D 2 denote the outside and inside diameters respec- 
tively. 

56. Combined Stresses 

In various cases combinations of stresses may exist, such, for 
example, as tension or compression combined with shear, tension 
or compression combined with bending, tension or compression 
combined with torsion, bending combined with torsion. 

In order to find the maximum intensity of stress in such cases 
recourse must be had to the principles and methods of mechanics 
as developed in textbooks on that subject. The principles of the 
present section will serve to develop the conditions of the pipe or 
pipe line element as to the magnitude and location of the loads. 
Beyond this point the problem becomes one of mechanics. 



CHAPTER V 

MATERIALS, CONSTRUCTION, DESIGN 

The main purpose of the present work is the discussion of the various 
hydraulic problems which may arise in connection with the trans- 
port of liquids through pipe lines. No attempt is made therefore 
to present, with any degree of fullness, discussion of constructive 
features. In fact, a presentation of the constructive features of 
pipe lines and their mountings and attachments, including valves, 
elbows, Y's, expansion joints, etc., and treated with reasonable 
fullness from the standpoint of description and general discussion, 
would require a volume in itself. 

The purpose of the present chapter is therefore to present briefly, 
and from the standpoint of description and general discussion, 
mention of the more important structural elements of pipe line 
construction, and with some reference to the more important 
problems which may arise in connection with them, but without 
attempt to approach a comprehensive treatment of this phase of 
the subject. 

57. Materials 

The materials commonly employed for the construction of pipe 
lines are steel, cast iron, wood stave and reinforced concrete. 
Steel is employed in two forms, plate or sheet steel and cast steel. 
Plate or sheet steel is employed for two classes of pipe as follows : 

(a) Commercial pipe as commonly employed for piping steam, 

air, gas, water, etc., and in sizes from -J- inch to 12 or 15 
inches inside diameter and up to 30 inches outside diameter. 

(b) Pipe made up of steel plates with longitudinal joint either 

riveted or welded, with diameter from 16 or 18 inches to 

7 or 8 feet, and with thickness of plate to suit the special 

requirements of the case. 

Cast steel is employed for bends, elbows, Y branches, flanges, 

saddles and other like connections or accessory mountings, and 

also occasionally for short connecting lengths of pipe. 

Cast iron is employed in the form of lengths (usually 12 feet), 
with the well-known bell and spigot form of joint, and in com- 
mercial sizes and thicknesses of metal up to 84 inches diameter, and 
in special sizes and thicknesses of metal to suit the requirements of 

190 



MATERIALS, CONSTRUCTION, DESIGN 191 

the case. Cast iron is also employed for bends, elbows, Y branches, 
flanges, saddles, and other like connections or accessory mountings, 
but where the pressure requirements are relatively light. Where 
the pressure requirements are moderate to high and in the best 
grade of work for all pressures except the very lowest, such items 
should be made of cast steel. 

Wood stave pipe with steel rod or wire band reinforce is employed 
for moderate pressures, and where the special characteristics of 
such pipe may meet the requirements of the situation. 

Reinforced concrete has been only sparingly employed for pipe 
line construction. With sufficient steel reinforce it may readily 
be made adequate in strength for moderate pressure requirements. 
Due, however, to the fact that the tensile strength of concrete is 
low and that full assurance cannot be had of the simultaneous 
development of stress in both concrete and steel each in proportion 
to its safe strength, it is necessary in practice to supply steel cir- 
cumferential reinforce sufficient to carry the full internal load by 
itself and without assuming aid from the concrete.' 

In the case of moderate to high pressures, therefore, where the 
pressure is the determining element with regard to thickness of 
wall, there would be no advantage in using concrete except for its 
value as a protective coating for the steel. In the case of light 
pressures, however, where the amount of steel required for strength 
is far less than that required for stiffness, resistance to external 
collapse, and for durability under corrosion, the combination of 
adequate reinforce steel for strength under internal pressure with 
concrete for stiffness and for protection, may present advantages 
sufficient to justify its use. 

Broadly speaking, however, steel or cast-iron pipe with like 
fittings and connections furnish the typical or representative 
practice in pipe line construction. Furthermore, while the general 
principles of pipe line flow are independent of size, many of the 
special problems considered in the present work imply the larger 
sizes of pipe line, such as would be typical of power plant practice 
or of the transport of large volumes of liquid over long distances. 

We may now consider in some further detail the principal types 
and forms of pipe and pipe line construction. 



58. Commercial Pipe 

This is placed on the market in three grades as regards strength, 
known respectively as standard, extra strong and double extra strong. 
Commercial sizes of standard pipe as shown on recent manufacturers' 
lists show nominal sizes ranging from £ inch to 15 inches rated 
by inside diameter and sizes ranging from 14 to 30 inches rated 
by outside diameter. The latter is commonly designated as O.D. 
pipe. For the pipe rated on inside diameter the actual diameter 



192 HYDRAULICS OF PIPE LINES 

differs somewhat from the rated diameter, usually in excess, especi- 
ally in the smaller sizes. The actual outer diameter of O.D. pipe 
agrees with the nominal rating. 

Standard engineering handbooks may be consulted for details 
regarding the characteristics and dimensions of such pipe. 

In the case of extra strong and double extra strong, the added 
thickness is placed on the inside, thus reducing the inside diameter, 
but leaving the outside diameter the same, and thus suited to the 
same fittings and screw connections as standard pipe. In so far 
as standard fittings and connections may be considered adequate, 
they may therefore be used with the extra and double extra strength 
pipe. If otherwise, special fittings and connections are required, 
the tapping size will in any event be the same as for standard sizes 
of pipe. 

Engineering handbooks or dealers' lists may be consulted for 
details regarding the characteristics and dimensions of such pipe. 




Fig. 109. — Bell and Spigot Joint. 

O.D. pipe is made as noted above in diameters listed by the 
manufacturers, from 14 to 30 inches, and in varying thickness 
according to service required, from J to 1 J inches. 

The usual connections and fittings for commercial pipe, such as 
bends, turns, elbows, tees, Y branches, valves, etc., are too well 
known to require detailed consideration in the present work. Full 
information regarding these matters is furthermore available in the 
various manufacturers' or dealers' lists. 



59. Cast-iron Pipe. Commercial Sizes and 
Standards 

Commerical cast-iron pipe is made in a wide variety of diameters 
and thicknesses of metal, in accordance with varying requirements 
and varying standards of design and manufacture. The nominal 
size is measured on the inside, and the varying thickness of metal 
affects therefore only the outside diameter. 

The usual manner of connecting successive lengths of such pipe 
is by means of the bell and spigot joint formed with space for calked 
metallic lead as packing material, as shown in Fig. 109. 

Engineering handbooks or dealers' lists may be consulted for 
details regarding the characteristics and dimensions of such pipe. 

In addition to the standard bell and spigot form of joint, various 



MATERIALS, CONSTRUCTION, DESIGN 



193 



forms of special joint are occasionally employed. Thus in Fig. 110 
are shown two forms of flexible joint, lead packed. The form 
shown at a is more commonly employed, while that shown at b 
is more expensive and is intended for large pipe under relatively 
high pressure. So-called " Universal" pipe, as shown in Fig. Ill, 




Jfc 



% 



VI 



V ul 

Fig. 110. — Special Forms Ball and; Socket Joints. 



is fitted with an inside and outside taper or cone joint with machined 
surfaces, giving an iron on iron contact. The slope of the tapers 
are slightly different, thus allowing some degree of flexibility while 
still remaining tight. The two parts of the joint are drawn together 
by bolts carried in lugs as shown. 

A great variety of formulae have been proposed and employed, 
giving the relation between the diameter, thickness and pressure 




Fig. 111. — Universal Pipe Joint. 



or head for safe operating conditions. Among these Fanning's, 
which has had, perhaps, as wide acceptance as any, may be put 
in the following form : 

7220 i ~ 3 
where t= thickness (i). 
| p — pressure (pi2 ) . 
d= diameter (i). 
h.p.l. — o 



194 HYDRAULICS OF PIPE LINES 

This formula, which gives results agreeing well with the figures 
of standard practice, implies a wide variation in the actual working 
stress for varying sizes and pressures. From the form of the ex- 
pression it is clear that the thickness t provides for the following 
combination : 

1. A working pressure p with a working stress of 3610 (pi2). 

2. An excess pressure of 97-2 (pi2) (or in round number 100 (pi2)) 

as the result of shock or other unusual conditions, and with 
the same working stress. 

3. An excess thickness of J inch to allow for corrosion and wear, 

accidents of manufacture, etc. 

In addition to the standard thickness of cast-iron pipe and 
fittings suited to meet ordinary requirements, specially heavy 
grades are listed by large manufacturers and intended to meet the 
requirements of specially high pressures, the thickness and other 
dimensions being graded to the working pressure to be carried, 
usually in 100 -pound steps up to 400 or 500 pounds per square 
inch. Manufacturers' lists may be consulted for the details of 
such extra heavy pipe. 



60. Sheet Steel Pipe 

Sheet steel pipe is made up in lengths according to the available 
dimensions of steel plate, and joined length to length either by 
circumferential riveted joints or by means of flanges or other special 
form of joint. Longitudinal joints are either riveted or welded. 
As a general rule each length or section of pipe is made up of a 
single plate wrapped around the circumference, and hence with 
but one longitudinal joint per section. 

Taking first the form with riveted joints we shall pass in rapid 
review the chief constructive features. 

The available materials are sheet steel plates in widths up to 
8 or 10 feet, in lengths up to about 20 feet, and in thicknesses 
increasing by sixteenths up to 1| inches or more if desired. 

It is shown in mechanics that in the case of a cylinder under 
internal pressure the stress along a longitudinal ideal section is 
twice that along a circumferential section. It follows that the 
chief effort, in the matter of fastenings, must be directed toward 
the development of the highest possible efficiency in the longitudinal 
riveted joint. The only exception to this general rule is found in 
the case of pipes subjected to light pressure where strength is not 
the ruling factor in determining the thickness. An illustration will 
make clear the considerations involved. Assume a diameter of 
24 inches and a pressure of 30 (pi2) with a safe actual working 
stress of 15,000 (pi2) and a longitudinal joint efficiency of 80 per 



MATERIALS, CONSTRUCTION, DESIGN 195 

cent. From mechanics we have for the thickness of a pipe under 
internal pressure, the formula : 

fd 
2Te 
where p= pressure (pi2). 
d= diameter (i). 
t= thickness (i). 
T=safe working stress in joint. 
e=joint efficiency. 

Substituting in this formula we find £=-03. 

It appears, then, that so far as strength alone is concerned a 
thickness of -03 or, say, ^ inch would be sufficient under these 
conditions. 

Such a pipe, however, would lack stiffness and resistance to 
external local forces or to collapse under external pressure in case 
the pressure within should ever fall below the atmosphere. Further- 
more, there is no provision against corrosion or wear, except as 
contained in the factor of safety. Thus with an ultimate strength 
of steel plate at 60,000 (pi2) the factor of safety when new is 4. 
Suppose now a thickness of no more than -01 inch to disappear 
under the influence of corrosive action. This means the loss of 
one -third the available metal and the reduction of the factor of 
safety to 2*67. With ^ inch thickness removed by corrosion, only 
one -half the original metal remains and the factor of safety has 
fallen to 2. The serious results of such a condition are plainly 
apparent, and in order to provide for a reasonable life under corro- 
sive action and also to give stiffness and strength under external 
load, it is necessary in all such cases to add arbitrarily to the thick- 
ness. For the case mentioned, £ or T ^ inch would be considered 
the minimum thickness permissible. For all such cases it appears, 
therefore, that strength under internal pressure is not the ruling 
factor in the determination of thickness, and hence that the realiza- 
tion of the highest possible efficiency in the longitudinal joint is of 
less importance than in cases where, under high pressures, strength 
under such pressure becomes the ruling factor. 

Longitudinal Joints. — For the longitudinal joints of steel plate 
pipe line sections, a great variety of design may be employed, 
according to the efficiency to be considered significant in the special 
case. 

Thus when stiffness and allowance for corrosion determine a 
thickness considerably in excess of the requirements for strength 
against internal pressure, a single riveted lap joint may be employed. 
A double riveted lap joint will give a somewhat higher efficiency and 
a joint more easily made and kept watertight. Next in order, and 
where it may be desirable to maintain a more truly circular form of 
section, comes the single or external butt strap joint with either 
single or double riveting, as indicated in Fig. 112. The efficiency of 



196 



HYDRAULICS OF PIPE LINES 



the single riveted lap joint will range about -55 or -56 and of the 
double riveted lap joint about *70. The efficiency of the single and 
double riveted single butt strap joints will range about the same as 
for the corresponding lap joints, the only advantage being in a more 
truly circular form of pipe section in the latter case. 

Where high efficiency is desired, as in all cases where thickness for 
strength against internal pressure is the determining feature, the 
lap or single butt strap joints should not be employed. For such 
cases double butt straps with the three or even four rows of riveting 
on each side are employed, raising the efficiency of the joint to from 
•80 to -90 or more according to the particular design of joint 
employed. 

A detailed discussion of the theory and manifold forms of riveted 
joints is outside the scope of the present work. In Figs. 113, 114, 
however, are shown diagrams of approved forms of such joints as 
may be applied to the longitudinal seams of steel pressure pipes. 




i O : O 




/ o J o 


/ 


! o:o 


( 


JO i O 


\ 




0;0 


\ 




o : o 






o;o 





Fig. 112. — Riveted Joints. 

In connection with these diagrams a few indications may be given 
regarding the examination of any proposed form of joint for strength 
and efficiency. 

It should first be noted that such procedure is in considerable 
degree arbitrary in character, since we do not know the influence of 
the friction between the plates and of various other factors which 
may affect the relation between the load and the manner in which it 
is carried by the component elements of the joint. With the usual 
conventions, however, the joint may be examined as follows. 

Obviously the resistance to rupture in all possible ways should be 
examined, and the strength for each such possible method com- 
pared with the strength of the plate as a whole. Actually only 
three possible modes of rupture are commonly considered. 

1. Rupture by tearing between the outer rivets of the rivet 

pattern. 

2. Rupture by shearing the rivet sections. 

3. Rupture by crushing due to the bearing load between the rivet 

and the metal of the plate. 



MATERIALS, CONSTRUCTION, DESIGN 



197 



The unit of the joint commonly taken is the element of the rivet 
pattern covering a distance equal to one space in the outer row of 
rivets, as AB, Figs. 113, 114. 

Let p=pitch of rivets in the outer row. 
£=thickness of plate. 
d= diameter of rivet. 




A B 

V - & ^.v.- & -::|v.-^-.-.,|-.-. Q - -■::■■ 

ooooiooioo " 
.oooopoooo 

16 oo b 6 o'ooo 
oooooooo 

-O O O O---- 



Fig. 113. — Riveted Joints for Longitudinal 

Seams. 




Fig. 114. — Riveted Joint fob Longitudinal 

Seams. 



198 HYDKAULICS OF PIPE LINES 

m=numberof rivets in single shear in one element of the 

rivet pattern. 
w=number of rivets in double shear in one element of the 

rivet pattern. 
T= tensile strength of plate. 

S 1 = shearing strengths of one section of rivet in single shear. 
8 2 = shearing strength of one section of rivet in double shear. 
G 1 = crushing strength for rivets in single shear. 
<7 2 = crushing strength for rivets in double shear. 
Then for the total strength of plate of width p we have 

(1) ptT. 

For the strength against tearing along the line between the two 
rivets in the outer row we have 

(2) ( V -d)tT. 

For the strength against rupture by shear we have 

(3) (mS 1 +2n8 2 ). 

For the strength against failure by crushing we have 

(4) dtimC^nCJ. 

The various efficiencies will be given by dividing the various 
expressions (2), (3), (4) by (1). 

Obviously the lowest value must be considered as ruling for the 
joint in question. 

The highest economy of joint is obtained when the proportions 
are such as to give equal values to all three efficiencies. This, 
however, is not always practica^e, though in most cases this 
condition may be closely realized. 

For steel plates and rivets the following values may be employed 
for the various strength factors above noted. 

T= 60,000 
S 1= = 44,000 
S 2 = 45,000 
<7 1= 90,000 
C 2 = 110,000 

The distance of the centre line of the row of rivets nearest the 
edge of the plate should be from 1-5 to 2 times the rivet diameter. 

The rivet diameter d should be from 1-2 t to 1-4 t. 

In multiple staggered riveted joints the minimum distance between 
rows of rivets should be -6 to -8 the minimum pitch. 

Welded Longitudinal Joints. — In place of riveted joints, welded 
longitudinal joints have, during the past decade, come into extended 
and approved use. Such joints are made lap -welded in gas -heated 
furnaces and with approved technique in the process show effi- 
ciencies of 90 per cent and better. Such form of joint is especially 
suited to the heavier thicknesses found necessary for the lower ends 
of high pressure penstocks and similar designs. Where the thickness 
would exceed 1 inch, the equivalent strength may be made up by a 
main shell with welded bands shrunken on, three or four inches 
wide and with an equal distance in the clear between bands. In 



MATERIALS, CONSTRUCTION, DESIGN 199 

this manner, pipe for the heaviest service may be made up in 
welded form. 

Welded pipe, due to the higher joint efficiency as compared with 
riveted pipe, has the advantage of thinner plates and less weight 
for the same diameter and head or greater diameter for the same 
thickness and head. Due to the absence of rivet heads it has also 
the advantage of better hydraulic conditions of flow, as referred to 
in Sec. 61. 

Circumferential Joints. — As already noted in the case of a 
cylindrical shell subject to internal pressure, the stress along a 
longitudinal line or section is twice that along a circumferential 
line or section. Or otherwise, a cylindrical shell under internal 
pressure has twice the strength against rupture along a circumfer- 
ential line that it has against rupture along a longitudinal line. It 
follows that with equal efficiency in both longitudinal and circum- 
ferential joints the factor of safety will be twice as great against 
rupture along a circumferential line as compared with rupture along 
a longitudinal line. It results that, so far as strength alone is con- 
cerned, there is no occasion for using the especially high efficiency 
riveted joints which are required for the longitudinal seams. The 
efficiency of a properly proportioned single riveted joint is usually 
found about 55 per cent. Such a joint in a circumferential seam would 
therefore represent a factor of safety equal to that for a longi- 
tudinal joint with efficiency of 110 per cent were such efficiency 
possible. Or otherwise, with the very best efficiency of longi- 
tudinal joint possible, even supposing it to reach 100 per cent, and 
with a single riveted circumferential joint of 55 per cent efficiency, 
there will still remain an excess factor of safety with regard to 
rupture along the circumferential joint, and if tested to destruction, 
rupture will occur along the longitudinal joint. 

It should be noted further that the development of full stress 
along a circumferential line in a cylinder under internal pressure 
presupposes a cylinder with closed ends and without external 
constraint. The condition of a pipe with open ends and through 
which is flowing a stream of water is far from fulfilling these 
specifications. In fact it is readily seen that in a typical pipe line, 
stress along a circumferential line will only be developed as a result 
of some combination of the following conditions and of which (4) 
must be a constituent element. 

(1) Bends, turns or elbows. 

(2) Changes in size. 

(3) Closed valves. 

(4) Such freedom from constraint at or near the conditions (1), 
(2) or (3) as to permit end movement relative to some other part of 
the pipe definitely anchored in place, thus developing a lengthwise 
pull on the pipe as a whole and a resultant stress along a circumfer- 
ential line. 

With actual pipe lines these conditions are likely to obtain in 



200 



HYDKAULICS OF PIPE LINES 



varying degrees, but it is unlikely that they will be such as to 
permit the development of the full lengthwise pull due to the 
internal pressure over a closed end of the pipe, and hence of the 
development of the full stress along a circumferential line (see 
Chap. IV, Sec. 52) ; 

All of these considerations show, so far as we are concerned with 
strength against internal pressure, the greater relative importance 
of high efficiency in the longitudinal as compared with the circum- 
ferential joints, and the sufficiency of a properly proportioned 
single riveted lap joint. 

The circumferential joint must, however, provide for other 
requirements quite independent of strength under internal pressure. 
These are : 

1. Strength funder bending stress to which the pipe may 
be subjectedjand general stiffness, coherence and continuity of 
strength. 

2. Watertight closure of the joint. While the single riveted joint 
can be made watertight under normal working conditions, it is 

much easier to make a double riveted 
joint tight, and the latter is much 
less liable to develop leaks under 
irregular shifting stresses due to 
changes in the temperature condi- 
tions. The double riveted joint is also 
markedly superior to the single riveted 
in stiffness and in giving strength to 
the pipe under bending stress. For 
these reasons the double riveted form 
of joint is to be recommended for 

the circumferential seams in all high-grade work. 

In connecting the successive sections of the line choice will lie 

between two modes of construction : 

1. In and out sections, or alternate sections differing in radius by 
an amount equal to the thickness of the plate, and thus adapted to 
form the lap circumferential joint by the slipping of the smaller 
section into the larger. 

2. Sections all of the same radius or diameter, butting together 
at the end and connected by an outside butt-strap, thus making a 
double lap joint (see Fig. 115). 

The use of in and out sections has the advantage of a slight saving 
in cost and of being easier to calk from the inside. It has, however, 
the disadvantage of a periodic change in pipe diameter at every 
section, alternately large and small, so that there will be at each 
joint a definite loss of head (see Sec. 9) due to this cause. The use 
of uniform sections with outside butt-strap avoids this loss and 
thus furnishes much better hydraulic conditions than* with alter- 
nating diameters. 

To realize this advantage, however, the space left for calking, if 




Fig. 115. — Riveted Joint fob, 
Circumferential Seams. 



MATERIALS, CONSTRUCTION, DESIGN 201 

any, between the ends of the sections must be filled in, otherwise 
there will be presumably little to choose between the two forms of 
pipe. We shall again refer to this subject at a later point. 

By reason of the possibility of better hydraulic conditions, the 
use of sections of uniform diameter and with outside butt -straps is 
to be recommended in high-grade practice. 

61. Hydraulic Conditions in Riveted Pipe Lines 

The features which may enter into the production of eddy losses 
in riveted pipe lines are the following : 

1. Rivet heads in both longitudinal and circumferential joints. 

2. Longitudinal butt-strap ends. 

3. Abrupt changes in size at the ends of sections, in case in and 
out sections are used. 

4. Abrupt enlargement in size between the ends of section of 
uniform diameter, in case the ends are not butted closely together, 
or in case the space left between the ends is not otherwise filled in. 




Fig. 116. — Riveted Joint foe, Circumferential 
Seams with Spun Lead Filling between 
Ends or Sections. 

In a pipe line of considerable length there may be thousands of 
rivet heads, each projecting a little way into the moving stream of 
water and producing an eddy of small individual magnitude, but in 
the aggregate forming a loss by no means negligible in amount. The 
possible magnitude of this loss and the means for reducing it to a 
minimum have not attracted the attention which they deserve. 
Instead of the full projecting head, as is too commonly employed, the 
use of a countersunk nearly flush head or point, as shown in Fig. 116, 
would be advantageous. This gives the effect of smooth ship plate 
riveting and provides improved hydraulic conditions as compared 
with the projecting head. Pipe rivets are commonly driven, heads 
inside and points outside. This is primarily as a matter of con- 
venience. With such mode of riveting, the rivet for the results of 
Fig. 116 must be formed with a special head. If, on the other hand, 
the rivets are driven head outside and point inside, the usual form of 
rivet will serve and it only remains to countersink the hole at the 
proper angle, head up, as in ship work, and chip off the excess. 
While the saving by such form of riveting can scarcely be estimated 
with any approach to accuracy, it seems well assured that it is by 
no means negligible in amount. 

Longitudinal butt -strap ends have already been referred to. In 



202 HYDKAULICS OF PIPE LINES 

common practice these are often shifted through a certain angle at 
each joint, usually 90°. This gives a series of ends, two for each 
section of pipe, all unshielded and all operating to produce loss 
through eddies and turbulence. There seems to be no necessary 
reason for shifting the straps. It cannot add essentially to the 
strength of the line. If then the straps are lined up along a single 
element of the cylindrical pipe, they produce simply a change in the 
form of cross section and doubtless introduce a slight loss along their 
longitudinal edges, but the loss due to their ends may be practically 
eliminated. 

The influence of the abrupt change in size at the ends of sections 
where the in and out system is used, has been referred to previously. 
While not directly measurable and only to be inferred by com- 
parison, the loss due to such ends is undoubtedly far from negligible. 
It may be obviated by the use of uniform sections with outside 
butt-straps, assuming proper care of the space between the ends as 
noted below. 

The influence due to the sudden enlargement between the ends of 
pipes fitted with joints, as in Fig. 115, has been already referred to. 
If this space is left open, a loss will result and the possible advantage 
of the uniform diameter of sections will be largely, if not wholly, 
lost. To avoid this, the space should either be filled with some 
material such as spun lead, or the joints may be trimmed and butted 
close with electric welding in lieu of calking, as referred to below. 
The advantage, in respect of the hydraulic conditions of flow, 
offered by welded longitudinal joints in place of riveted joints, has 
already been noted in Sec. 60. 

62. Calking of Riveted Seams 

The calking of a riveted seam is much the more effective when 
applied on the side under pressure. It is obvious that it is easier 
to stop a small leak in a seam on the entering side rather than on the 
issuing side. Hence in pipe work inside calking is much more 
effective than outside calking. On the other hand, with the pipe 
under pressure, local leaks or imperfect sections of the joint can only 
be closed from the outside. For these reasons it may be recom- 
mended to carefully and thoroughly calk all seams on the inside 
before applying the pressure. Then for the closure of such small 
residual leaks as may show, or for the closure of small leaks which 
may develop with the pipe in service external calking may be 
resorted to. 

In this connection reference may be made to the circumferential 
joint formed by an outside butt-strap, as shown in Fig. 116. If the 
section ends are butted close together it will not be practicable to 
calk on the inside at the angles a and b. On the other hand, unless 
the plates are trimmed with great nicety, they will not butt together 
closely all the way around, and a crack or opening will be left 



MATERIALS, CONSTRUCTION, DESIGN 203 

between the plates of varying width. To permit of calking at a and 
b the plate ends may be separated a distance somewhat less than 
the thickness, as shown in the figure. A special tool may then be 
employed for calking at a and b, thus insuring, as nearly as may be, 
a watertight closure of the joint. 

This, however, leaves an opening between the plates sufficient to 
form an eddy of sensible magnitude as a result of the sudden 
enlargement in size of conduit. This again may be closed by calking 
spun lead into the opening. This is metallic lead in the form of fine 
threads, and forms an admirable material for filling such spaces. 
If the edges of the plates are slightly undercut, it will aid in holding 
this in place. In any event, the undercut developed by the calking 
at a and b will operate to form an anchor at the bottom for such lead 
filling. In this manner all requirements may be met ; the joint 
may be calked on the inside and a smooth continuous surface 
provided for good flow conditions for the water. 



63. Electric Welding at the Joints in Lieu of 

Calking 

Modern developments in the art of electric welding with metal 
electrodes furnish an admirable substitute for the time-honoured 
practice of impact calking. By this means the entire line of contact 
between the two plates forming the joint may be closed and the 
plates locally united by welded metal filling. The use of local 
electric welding instead of calking may be strongly urged in all high- 
grade practice. It unquestionably gives the nearest approach to 
absolute assurance of a watertight joint. There is presumably less 
choice than with calking as to application on the inside or outside. 
Application on either side, as may be convenient, will be effective, 
or if extra assurance is desired, both inside and outside seams may be 
treated. A further advantage of electric welding in lieu of calking 
is realized with circumferential joints, as in Fig. 116. Here the 
seams at a and b are readily treated, or otherwise the electric calking 
may be restricted to the outside seams at c, d. 

Welding by the oxy-acetylene process may also be applied to the 
same end, but it is somewhat less convenient in application for this 
particular purpose, and it will also be usually found somewhat more 
expensive in use. 

64. Pipe Joints and Connections 

The connection of the successive lengths of riveted pipe by means 
of circumferential riveted joints has already been referred to in 
Sec. 60. When the pipe is to withstand high pressure, as at the 
lower end of a high head line, or at the pump end of a high-pressure 
pumping line, some form of flange joint is often employed. In the 



204 



HYDKAULICS OF PIPE LINES 



case of welded pipe, a number of special forms of joint have been 
developed in addition to the regular type of flange connection. 

Fig. 117a shows a standard form of flange joint in which the 
flanges are riveted to the sections of pipe. 

In Fig. 1176 the flanges are likewise riveted to the pipe, but the 
form of packing is peculiar and of special value for very high 
pressures. As shown in the figure, a groove of section narrowing 




Fig. 117. — Forms or Flange Joints. 

toward the outside is formed in one of the flanges. Within this 
groove is fitted a ring of soft, solid rubber packing. The water has 
free access to the base of this groove and the pressure acts on the 
rubber ring tending to drive it more and more closely into the 
narrow end of the groove, thus effectually packing the joint even 
against the very highest pressures employed. This form of packing 
joint is also employed with entire success against pressures of 




Fig. 118. — Forms of Flange Joints. 



several thousand pounds per square inch — far above any pressures 
liable to be met with in pipe line practice. 

In Fig. 118a the ends of the sections of pipe are flanged out and 
are pinched together between specially formed ring flanges as shown. 
When the nuts are properly set up on the through bolts holding the 
two parts of the flange together, the projecting rib on the outer edge 
insures a definite compression on the flanges and on the packing 
material between them. The individual rings in the case of such a 
construction must be made in two parts in order to get them into 
place back of the flange on the pipe. 



MATEKIALS, CONSTRUCTION, DESIGN 



205 



Fig. 1186 shows a favourite form of joint for thin and medium 
thickness of large welded pipe. 

Fig. 119a shows a form of connection which gives in effect an 
expansion joint at each joint of the line. The construction will be 
clear from the diagram. 

Fig. 1196 shows an excellent though somewhat expensive form of 




Fig. 119. — Forms of Flange Joints. 

joint for high-pressure work. The ends of the pipe sections are 
thickened up by special treatment in fabrication, and are turned up 
to form as shown. Within the thickened end is formed a tapering 
groove for soft rubber ring packing, similar to that in Fig. 1176. 
The action of the flange rings on the sloping surfaces formed on the 
pipe is evident from the diagram. 

Fig. 120 shows a form of joint suited to very heavy pipe. Here 




Fig. 120. — Form of Flange Joint. 



the flanges are threaded on to the pipe, and a groove or recess for 
soft rubber ring packing is formed in the metal of the pipe itself 
without thickening. 



65. Wood Stave Pipe 

See Fig. 121. This pipe is made in two forms, machine banded in 
lengths, and continuous. Machine -banded pipe may range from 2 
or 3 inches up to 24 inches in diameter and in lengths up to 20 or 
24 feet. It is banded with heavy wire wound on by machine under 
appropriate tension. For connecting together successive lengths, 
different types of coupling may be employed. 

In one type of such coupling a coupling band is used, the inside 



206 



HYDRAULICS OF PIPE LINES 



diameter of which is only slightly less than the outside diameter of 
the pipe. The ends of the pipe are then slightly reduced in diameter 
so that a tight joint may be made with the coupling band. The 
latter is then forced for half its length on to the end of one of the 
lengths of pipe and the next length is forced into the other side of the 

coupling, thus completing the joint. A 
coupling band of this character is made 
up the same as a short length of pipe with 
wire or steel -rod banding to give it the 
necessary strength. In fact such coupling 
bands are usually banded up to perhaps 
double the strength of the pipe, in order 
that they may stand the stress which 
develops from forcing them on to the end 
of the pipe. 

A second form of coupling shows a 

similar band or sleeve, but with inside 

diameter somewhat less than the outside 

diameter of pipe. The latter at the ends is then reduced in 

diameter to two-thirds or one-half thickness of the wood for the 

length of bearing, and the joint is assembled as before. 




Fig. 121.— Section of 
Wood Stave Pipe. 




Fig. 122. — Saddle for Round Rod Ties — Wood Stave Pipe. 



In a third form, used only for small pipes and light pressures, 
one end of a pipe length is reduced in diameter to about half the 
thickness of the wood and a corresponding counter bore is made in 
the companion end of the adjacent length. These are then forced 
together and the joint is complete. 

Instead of wood couplings, cast-iron couplings of suitable form are 
sometimes employed. For elbows, Ys and Tees, cast-iron fittings 
are commonly employed. 

Continuous wood-stave pipe is made by assemblage in place, 
breaking joints with the successive staves so that the entire line 
becomes a continuous structure without joint or break in con- 
tinuity. The banding of such pipe consists of separate ring bands, 
usually of. round steel, threaded at one or both ends, fitted with a 
suitable saddle or end fitting and set up with nut, as shown in 
Fig. 122. Such pipe is made in diameters ranging from 12 inches to 
100 inches and more. 



MATERIALS, CONSTRUCTION, DESIGN 



207 



The following thicknesses of staves for machine -banded pipe have 
been generally adopted by manufacturers of this style of pipe. 



Diameter. 


Thickness. 


3 inch. 


1 inch. 


4,5,6 „ 


ItV ., 


8,10 „ 


li „ 


12,14 „ 


itV .. 


16,18,20 „ 


ii „ 


22,24 „ 


1A .. 



For continuous stave pipe the thickness of the stave increases 
somewhat with the size and with the pressure. The variation with 
the pressure has two purposes : 

1. To secure increased general rigidity and solidity of construc- 

tion with advancing pressures. 

2. To allow for the necessary bearing pressure between the sides 

of the staves. 
The following table gives the general range of thicknesses 
employed : 



Diameter, 
Under 25 inches. 
24-26 „ 
28-34 „ 
36-44 „ 
46-54 „ 
56-72 „ 
74 upward 



Thickness. 
If inches. 

1* » 

li „ 

1* » 

If „ 

2 inches to 3 J inches 

2f „ 3f „ 



In pipe of this construction the band, when the pipe is under load, 
must carry the total load due to the internal pressure plus the 
reaction due to the compression between adjacent staves. Some 
degree of such compression is necessary in order to insure a water- 
tight joint with the pipe under load. The bands must therefore be 
put on with such initial tension as will insure the necessary degree 
of residual compression when under full load pressure. When the 
pipe is not under load the tension in the band will be that due to the 
original tension with which it is set up. When pressure is brought 
on the pipe the tendency will be to force the staves outward, 
relieving the compression but increasing the tension in the bands so 
that under actual working conditions the total band tension must 
equal, as above noted, the sum of the load due to water pressure 
plus the reaction due to the compression. 

Mr. A. L. Adams,* in recognition of this general principle, has 
developed a formula connecting the pressure, size and spacing of 
bands, thickness of stave, etc. He assumes the extra load due to 
edge compression in the staves to be measured by 1-5 times the 
water pressure over the contact area of the staves plus an allowance 

* " Trans. Am. Soc. C.E., 1899," p. 27. 



208 HYDRAULICS OF PIPE LINES 

of 100 (pi2) for swelling of the wood when wet. On this assumption 
the formula develops as follows : 

/= * 

J (R+l-5t)p+100t 
Where /=band spacing (i). 

#=tensile load carried by band (p), 
i?=internal radius of pipe (i). 
£=thickness of stave (i). 
p= water pressure (pi2). 

Mr. D. C. Henny* considers that the allowance of 100 (pi2) in the 
Adams formula for the swelling of wood when wet is unnecessary, 
and prefers the formula 

J (R+h5t)p 

The bands, when in the separate or ring form, as for continuous 
pipe, are upset at the thread ends so as to give full section of metal 
at the root of the thread. 

The working stress in the band may be taken from 12000 to 
15000 (pi2) according to character of service. 

In addition to the tensile load carried by the band itself due 
consideration must be given to the question of the bearing or 
crushing load on the wood under the band. For redwood (Sequoia 
sempevirens) a bearing value of 700 (pi2) may be employed. For 
Douglas fir (Pseudotsuga Mucronata) bearing values of 800 to 
1000 (pi2) are employed. In determining this value it is assumed 
that one -half the diameter of the round rod or wire bears against 
the wood. On this assumption and denoting the bearing pressure 
by B and diameter of band by d, we have for the relation between 
B and S the equation 

In selecting the banding there are two variables, the spacing of the 
bands and the diameter of the rod. If there were no limitations to 
either of these, values could always be selected which would give 
any desired combination of values of tensile and bearing stress. 
Both dimensions must, however, be limited. The spacing cannot be 
greater than 8 or 9 inches and only then for very low pressures. 
Again, bands smaller than J to | inch corrode too rapidly, while 
those greater than § inch are too stiff and difficult to readily handle. 
Within these limits the desired dimension will not always secure 
both bearing value and tensile stress as desired and hence it may 
become necessary to design for both and take the final dimensions 
according to which of the two controls. 

For the smaller sizes of pipe, for example 12-inch diameter and 
less, the bearing value is likely to be the controlling feature and 

* Ibid., p. 68. 



MATERIALS, CONSTRUCTION, DESIGN 



209 



must therefore be carefully considered. For the larger sizes of pipe 
the stress due to the pressure is usually found to rule and the bearing 
pressure may therefore be neglected in the computation. Where any 
doubt may exist, however, the determination should be made in both 
ways and the safer value taken. 

Table XXV shows the size and spacing of bands recommended 

TABLE XXV 



og 










BANDS 










8 -a 

is a 


a> 

s 
5 


Spacing in Inches for Heads in Feet 


.5 ft 


25 feet 


50 feet 


75 feet 


1C0 feet 


125 feet 


150 feet 


175 feet 


200 feet 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


10 


f 


9^ 


6| 


51 


A 1 
*T6 


Q 7 

°T6 


915 

Z T6 


2| 


2 T 5 6 


12 


t 


8| 


K15 

°16^ 


*A 


on 

^T6 


31 


91 1 

Z T6 


2| 


21 


14 


3 

8 


711 

'T6 


5| 


*A 


Q 5 

d T6 


2| 


93 

Z 8 


21 


11 


16 


t 


?A 


m 


3| 


3 


21 


9 3 
Z T6 


11 


m 


18 


3 
8 


6| 


±t 


Q 5 
d T6 


*tt 


21 


1 15 
1 T6 


m 


11 


20 


t 


H 


±1 


31 


21 


2^ 


113 

X 16 


1 9 

1 T6 


it 


22 


t 


51 


3M 


2M 


2f 


l*f 


m 


iA 


L 16 


24 


1 


Q 3 

y i6 


6 


41 


Q 9 
d T6 


2*1 


2i 


21 


ll 


27 


h 


8 T6 


5| 


41 


31 ! 2| 


2ft 


2 


lit 


30 


1 


715 
'16 


61 


3| 


3 


v 2-A 


2* 


113 
x 16 


u 


33 


1 


m 


413 

*16 


31 


2| 


21 


115 


lii 

A 16 


H 


36 


1 


?A 


4| 


3i 


2A 


21 


1^4 


ly 9 6 


1^ 

A 8 


39 


1 


6H 


*i 


01 


2^ 


2i 


1« 


11 


1 5*1 
L TSJ 


42 


1 


H 


4 


2. ->' 


n 


1* 


1* 


H 


11 


48 


1 


H 


org 


2 ¥ 


2 


1« 


lA 


11 


■Wl 


54 


f 


9f 


Kll 


4x6" 


Q*3 

d r6" 


2f 


2A 


115 


1» 


60 


f 


7| 


4| 


3f 


«tt 


21 


11 


If 


iA 


66 


1 


6} 


*A 


31 


2ft 


2 


If 


11 


i* 


72 


1 


°T6 


*A 


m 


9:5 


14 


1^ 


If 


U 


84' 


1 


6 


3f 


2| 


2 


If 


1 7 


11 


iA 


1 


8| 


51 


03 

«*4 


21 


2| 


2 


If 


1 9 


96 


f 


H 


31 


*ft 


If 


*ft 


11 


1* 




1 


7 


4f 


31 


21 


2 


If 


11 


x 16 


108 


3. 

4 


6f 


4 


21 


81 


11 


1 9 


If 


ift 


120 


I 


«A 


Oil 
^T6 


2f 


2^ 


1» 


1 7 

i T6- 


11 


1ft 



Band Spacing for Wood Stave Pipe, 



for service under various pressures as indicated in terms of water 
head.* 

It thus appears that in such a combination of wood staves and 
steel bands the latter provide the necessary strength under internal 
pressure while the staves provide for a watertight inclosure, insure 
the necessary stiffness and general coherence as a structure, at the 
same time protecting the bands against corrosion by the water. 



Furnished by Mr. J. D. Gallowayi 



[.P.L. 



210 HYDKAULICS OF PIPE LINES 

As already noted in Sec. 59, it is not practicable to reduce the 
thickness of steel pipe below some minimum value which according 
to size may range from J to perhaps t 5 q- or f inch. This minimum 
thickness is necessary regardless of stress due to internal pressure 
and in order to secure stiffness, resistance against collapse under 
external pressure and by way of excess material to provide adequate 
life under corrosion extending over a period of years. 

It thus results that under appropriate conditions, a combination 
of wood staves and steel bands may provide advantageous con- 
struction for light or medium pressures. 

By way of example assume a pipe 24 inches diameter under a 
pressure of 50 feet of water. Then from Table XXV we find per foot 
of length two J-inch bands giving a net section of -40 (i2) per foot 
of pipe. 

On the other hand, for an all-steel pipe under these conditions 
J to fV mcn would be considered a minimum thickness, at least 
having in view a reasonable life. This would mean a cross section 
of 1-5 to 2-25 (i2) of steel per foot of length for the all-steel pipe. 

It therefore appears in this case that strength alone can be 
secured by the use of one -fifth the amount of steel which an all- 
steel pipe would require. It then remains to be determined whether 
the combination per foot of -40 (i2) of steel section with the necessary 
wood staves will furnish a more economical or more satisfactory 
pipe than the use of 1*5 to 2-25 (i2) of steel section per foot as re- 
quired by all-steel pipe. This will naturally depend on special or 
local conditions and on the special advantages which the wood pipe 
may be able to offer. 

Among others the following special features may be noted. 

Wood stave pipe is not subject to corrosion or tuberculation, as 
in the case of iron or steel. It furthermore resists corrosion or 
damage under the attack of acidulated water or many chemicals 
which cannot be safely handled in iron or steel pipe. 

Regarding frictional resistance, the normal coefficients are 
somewhat better than for steel pipe. As noted in Sec. 7, when new, 
a value of n=-0ll may be employed, or values of C varying from 
120 to 135 according to size. Due also to the absence of corrosion 
the coefficients remain higher than for iron or steel pipe of the 
same age. The ultimate state of the interior surfaces of wood 
stave pipe, in common with many forms of water conduit tends 
toward the development of a sort of gelatinous slime covering 
with a value of n perhaps -012 or -013 and with values of C increased 
accordingly. 

Due to the elasticity of the wood and of the banding it is not 
liable to rupture when the contained water is frozen. The necessary 
expansion is taken up by the wood and by the banding. 

In the matter of durability, wood stave pipe of California red- 
wood (Sequoia sempervirens) has often given results comparing 
favourably with wrought iron or steel. It should, however, be kept 



MATERIALS, CONSTRUCTION, DESIGN 211 

filled with water. Alternate wetting and drying out of wood stave 
pipe will bring about rapid deterioration. Wood stave pipe, as a 
rule, decays from the outside, and it is therefore desirable to have 
such pipe exposed where it can be examined and kept painted with 
a suitable protective covering. The usual coating is hot tar with 
an outer coating of sawdust. If this coating can be kept intact 
and the pipe kept filled with water the life should range to twenty 
years and upward. 

The steel wire or rod banding must also be protected from cor- 
rosion by suitable protective coating, or otherwise the security of 
the pipe against rupture will be endangered. 

66. Reinforced Concrete Pipe 

Reinforced concrete pipe has already been briefly mentioned. 
The field of usefulness is restricted to light pressures, as for wood 
stave pipe, and for the same reason. It is, in fact, clearly seen 
that the concrete in the one case exercises the same function 
that the wood stave does in the other ; it supplies stiffness and 
protection against corrosion. 

In addition to circumferential reinforce, a certain amount of longi- 
tudinal reinforce must also be provided in order to give longitudinal 
coherence and strength, together with the needed strength under 
possible bending stresses. 

One serious limitation of concrete pipe is in the difficulty of 
making it watertight, especially under any but the lighter pressures. 
The consideration of ways and means for rendering concrete water- 
tight, or as nearly so as possible, is aside from the main purpose 
of the present work and the point can only be noted here in passing. 
The fact, however, must not be forgotten, and the difficulty of an 
entirely satisfactory realization of watertight concrete and of the 
avoidance of small cracks and consequent leakage must be con- 
sidered as constituting a serious limitation to the practicable use of 
such materials in pipe line construction. 

67. Design 

The given quantities in the usual problem of design are the 
following : 

1. The rate of flow or quantity of water to be handled. 

2. The head under which the line is to operate, and the general 

profile and topography of the line. 

The principal quantities to be determined are the following : 

1. The diameter or the general distribution of diameter along 

the length of the line. 

2. The determination of thickness and its distribution along 

the length of the line. 



212 HYDRAULICS OF PIPE LINES 

3. Details of joints and connections. 

4. Design of mountings and fixtures. 

5. Design of piers and anchors. 

Diameter of Line — Economic Size. — The determination of the 
diameter of a pipe line is an economic problem. Whether the line 
is to be used for the delivery of power to a water wheel or for the 
carrying of water or other liquid under a pressure head, the following 
fundamental elements are involved : 

(1) The lost head resulting from the flow through the pipe and 

the annual value of the head thus used up. 

(2) The size and weight of the pipe, the cost of the same installed 

in place and the annual charges on such cost. 

It is obvious that as the pipe is larger, item (1) will decrease and 
item (2) increase ; and inversely as the pipe is smaller. 

Further, it is evident that each of these items must be viewed in 
the light of an expense or cost and that the sum of the two repre- 
sents the total annual cost chargeable against the pipe line itself. 

The economic problem therefore reduces itself to the rinding of 
such a size of pipe as shall make the sum of these two items a 
minimum. In Appendix IV will be found a brief mathematical 
discussion of the general problem of the economic determination 
of a variable element in an engineering design. 

In the case of a pipe line under specially restricted and simple 
conditions it is possible to express algebraically the various 
quantities involved, and thus to derive an algebraic formula for 
the economical diameter in terms of the ruling conditions of the 
problem. It will be instructive to examine briefly this special case, 
in particular for the light which it sheds on the general character 
of the relation between the economic diameter and the controlling 
elements of the problem. The special conditions of the assumed 
case are as follows : 

1. The diameter is uniform. 

2. The gradient is uniform. 

3. The thickness bears a constant relation to the product of the 

pressure by the diameter. 

4. The weight per lineal foot bears a constant relation to the 

product of the thickness by the diameter. 

In the typical pipe line problem none of these conditions is ful- 
filled. The diameter commonly decreases in steps from upper to 
lower end, and for reasons to which further reference will be made 
at a later point. The gradient is rarely uniform, but more commonly 
follows, in some measure, the accidents of the topography. The 
thickness, instead of showing a uniform relation to the product of 
pressure and diameter (that is to the stress due to internal pressure), 
varies necessarily by steps according to the thickness of plates 
commercially available, and at the upper end, as we have previously 



MATERIALS, CONSTRUCTION, DESIGN 



213 



seen, there is likely to be a thickness in excess of that required for 
strength alone. 

In spite of these differences between the actual and the assumed 
case, the indications of the latter are highly instructive, and may 
serve further to give a very close first approximation in many actual 
cases. 

Letf X= annual charges on investment in pipe. 

Y— annual value of the head used up in friction. 

a = cost of pipe per pound installed in place (dollars). 

6=rate of interest for fixed charges on pipe (decimal). 

c= value of one horse -power year (dollars) estimated at 

lower end of pipe line. 
w= density of water (pf3). 
g 1 = density of steel (pf3), with allowance for laps, butt 

straps, rivet heads, etc. 
D= diameter of pipe (f). 
e=efficiency of longitudinal joint. 
I 7 = safe working stress in longitudinal joint (pi2). 
C= coefficient in Chezy formula. 
F=rate of flow (f3s). 
L=length of line (f). 
H and H as in Fig. 123. 




Fig. 123. — Design or Thickness for Shell of Pipe Line. 



Case 1. — For the above -indicated simple case we then readily find 
as follows : 

Thickness at lower end =-jrm — 7inr(f) 

2eT x 144 

Thickness at upper end= 2e y x °144 (f * 

Meantmckness^ ^f+ 2 f;f (f) 
4eJ'xl44 v ; 

Circumference = -jtD 

Hence X= ^^ 



(5) 



214 HYDRAULICS OF PIPE LINES 

64:V 2 L 

Lost Head= - omr .. (see Sec. 6) 

7T 2 C 2 £) 5 

„ 64cwF 3 L , m 

HenCC r= 550^O^ (6) 

Put P= ™°i( H + 2H o)L 



S= 



576eT 
6±wV*L 



550tt 2 O 2 
We then have X=abPD* (7) 

cS 



Y=^ 5 (») 



Then ^=2abPD 
dD 

dY_ 5cS 

dD~ D 6 
To find the conditions for a minimum of the sum of X and Y we 
put the sum of the derivatives equal to zero and solve for D. This 
gives : 

HSf- « 

Restoring the values of S and P, and collecting all numerical 
values into one term we have 

/ ecTV 3 \t 
D - 1 - 2726 (a60^ 1 (g+2 go) ) < 10 > 

The value of o - ! is intended to allow for the excess of weight of 
the actual pipe over that of a shell with geometrical volume ttDIL. 
The value to be used will therefore exceed the weight of a cubic 
foot of steel in the same ratio as the weight of an actual foot of pipe 
exceeds that of a volume irDt. 

Tables of weights based on good design show that for lap -riveted 
pipe the value of a x will be about 600, while for butt -strap pipe it 
will rise to values about 700, relatively larger in each case for thin 
plates and smaller for thick plates. More accurate values for any 
given case may be derived from the weights in Tables XXVI 
and XXVII. 

In this same connection, Merriman* gives a formula for the 
weight of lap-riveted pipe as follows : 

w=12-5Dt+10. 

Where w= weight in pounds per lineal foot and D and t are taken 
in inches. 

The indications of this formula agree fairly with the weights in 
Table XXVI. 

* " Civil Engineer's Hand Book." 



MATERIALS, CONSTRUCTION, DESIGN 215 

The form of (10) shows that D varies directly with eTc and V, 
and inversely with a, b, G and (H-\-2H ). The physical interpre- 
tation of these relations will prove interesting to the reader. We 
also note that the variation with all of the factors except V is very 
slow. With V it is a little less than as the square root. Thus a 
change of, say, 20 per cent in any of the factors except V will affect 
D by a little more than 2 J per cent. The same percentage change in V 
will affect D by about 8 per cent. The variation of D with a change 
in the various factors in (10) except V is then very sluggish or, in 
other words, any given value of D will hold approximately over a very 
considerable range of variation in these various factors. To change 
in V, the value of D is more quickly responsive as noted. It may 
also be noted that equal percentage changes in certain of the factors 
will leave the economic diameter unchanged. Thus equal changes 
in a and c will not affect the value of D. 

It may furthermore be readily shown by substitution from 
(9) in (7) and (8) that when the economic value of D is 
employed we shall have X=2-5F. That is, the total fixed 
charges on the pipe will be 2-5 times the annual value of the 
total lost head. 

Equation (10) thus serves to establish the economic value of a 
constant D on the assumption of a value of t varying and at all 
points proportional to a head varying from H to (H -\-H). An 
ideal line thus determined, would therefore extend over this range 
of head, with continuous change in t from one end to the other, 
while remaining fixed in diameter as determined in (10). 

It will be noted that the value of D thus found is independent of 
L, this term occurring on both sides of the final equation for D 
and thus cancelling out. Otherwise we note that both X and Y 
necessarily vary, each directly with L and hence the relation which 
makes their sum a maximum or minimum is independent of L. 
This means, in effect, that the value of D thus determined is inde- 
pendent of the line gradient. 

The line must, however, extend from head H to head (H -\-H), 
and hence the minimum length must be H, implying a vertical line 
With any other gradient the length will be greater, but no matter 
what the gradient may be (assumed constant) the value of D will 
give the economic diameter for a pipe of uniform size extending 
between these limits of head and with t continuously varying with 
the total head at the given point. 

In the case of a varying gradient, the total line may usually be 
divided into a series of parts, each with a sensibly uniform gradient. 
Then by the use of the same equation (10) the economic diameter 
for each section or part may be found, using for H the total head 
at the upper end of the section and for H the difference of head 
covered by the section. This will, in general, give as many different 
diameters as there are sections, continuously greater as we go from 
the lower to the upper end of the line. 



216 HYDEAULICS OF PIPE LINES 

Case 2.— Tapering Pipe Line with t continuously proportional to 

D(H -\-H). The assumption of a uniform diameter of line from 
one end to the other necessarily implies a limitation in the search 
for the most economic of all lines. The resulting D will indeed 
be the economic value on this assumption, but it does not follow 
that the removal of this restriction may not result in an economic 
result of still higher value. 

To develop the possibilities of a varying diameter let us seek 
to determine the economic value for an element of the pipe extending 
from total head (H +H) to (H +H-\- AH), where AH is^smal 
or (at the limit) differential element of head. 

In equation (10) the part played hjH here becomes (H -\-H), 
while H becomes AH. At the limit then (H-{-2H ) becomes 
2(H -{-H), and we have 

J-l-2726 ( a , !1T , m V (11) 



{pabaiCHHo+H) 



In this equation, H is, of course, to be taken as a variable with 
values ranging from to the value as in Fig. 123 at the bottom 
of the line ; or otherwise (H -\-H) is to be taken as the total head 
at the given point, ranging from one end of the line to the other. 

It is readily seen that this equation will give, from one end of 
the line to the other and for corresponding values of the total 
head (H -\-H), a series of values of D, increasing from bottom to 
top. The value at the bottom of the line where (H -\-H) is the total 
head, as in Fig. 123, will be less than the value given by equation 
(10) ; while the value at the top, where (H -{-H) becomes H , will 
be greater than the value by equation (10). 

By suitable investigation it may be shown that, for a section of 
uniform gradient, the ratio of the weights in the two cases is given 
as follows : 

^2 (H +H)V-H o * 

Wr H(H+2H o y< U2) 

Where W 2 — weight of tapered pipe 
W x = weight of uniform pipe. 

By a suitable investigation, it may also be shown that the ratio 
of the values of the friction head in the two cases, h 2 and ~h x is the 
same as for the weights. 

It may also be readily seen that with these values, W 2 and h 2 
will each be less, respectively, than W x and h x , their values becoming 
equal when H=0, that is, when the line becomes horizontal and 
in operation under the head H . 

Equation (11) gives therefore, in the general case, an economic 
value of D corresponding to each value of the total head (H -\-H) 
from top to bottom of line. With a profile of the line laid down, 
the distribution of D along the length is readily determined, and 
thus all characteristics of the line become known. 



MATERIALS, CONSTRUCTION, DESIGN 217 

Case 3. — Pipe Line with Stepwise Variation in Thickness.— 

Actual pipe lines cannot be made with thickness of shell varying 
continuously with the head or with the product of head and diame- 
ter. They must rather be made with thickness varying in steps in 
accordance with the commercial materials available. Steel plate is 
usually obtainable in steps of one-sixteenth inch, and assuming 
such or similar steps it becomes of interest to examine the problem 
of the economic diameter of line with such a stepwise distribution 
of thickness. 

For any one element or section of the line let 

£=thickness of shell (feet; 

AL=length. of section with constant thickness t (feet)j 

We have then, using the same general notation as before : 
X=abirDta 1 AL. 

While Y has the same value as before with the substitution of 
AL for L. 

Treating these in the same manner as before for the minimum 
value of I-f 7 we find 



/ 32cwV 3 \l 
D= \55^C 2 ab<j 1 t) < 13) 



or with w=62-4 

/ cF 3 \* 
D = l w(rid^t) < 14 ) 

It will be again noted that the value of D is independent of the 
element of length AL and for the same reason as in equation (10). 
It results that the value of D thus found is the economic value for 
this constant value of t, and from the lowest point where the head 
and thickness sustain the proper relation for strength, upward as 
far as this value of t is carried. 

Suppose then that t l9 t 2 , t 3 > etc., denote a series of thicknesses — 
differing, for example, by T \ inch or T ± Y f°°t- I^t D v D 2 , D 3 , etc., 
denote the corresponding values of D found by (14). Then the 
lowest point at which any given combination of D and t will fulfil 
the conditions for strength will be given by the relation : 

wD(H+H )=2%%etT (15) 

There will thus result a series of values of H which we may 
denote by H l9 H 2 , H 3 , etc. Then t x being the thinnest plate, it 
appears that the successive sections of pipe will extend over ranges 
of head as follows : 

D l9 t x from H=H 1 to If =0 
Z) 2 , t 2 from H=H 2 to H=H 1 
D s , t 3 from H=H 3 to H=H 2 

If for any reason t 2 or t 3 were taken as the minimum thickness, then 
similarly the combination D 2i t 2 or D z , t z would extend from H=H 2 
or H 3 to H=0. Otherwise it appears that beginning at the point 
where H=H 3 , for example, the combination D 3 , t 3 will extend 



218 HYDRAULICS OF PIPE LINES 

upward as far as the thickness t 3 is carried. Naturally this will 
be only until the next combination D 2 , t 2 can be made available. 
In this manner the range of head is determined for each combination 
of D and t. 

With the profile of the line laid down, the lengths of section 
corresponding to these various ranges of head H 1} (H 2 —H^), etc., 
are readily determined, and thus the line becomes known in all its 
characteristics. 

Influence of Variable Flow on Economic Size.— In the various 
cases of economic design thus far considered it is noted that the 
lost power is proportional to V 3 , the cube of the rate of volume 
flow. Hence in the case of variable flow the mean loss will be 
proportional to the mean cube of the variable rate of flow, or 
approximately to the mean cube of the variable power developed. 

If, therefore, we have given a curve showing the typical or 
accepted variation of flow through any unit period of time, as 
per day or week or month or year, we may cube the various ordinates 
of such a curve and find in any convenient manner the mean cube. 
This taken for V 3 in the preceding formulae will then give the con- 
ditions for economic design. 

General Comment on Economic Formulae. — With reference to 
the various formulae developed above for the economic diameter of 
a pipe line, there will result, in most cases, certain departures 
from any mathematical ideal as contemplated by a formula. These 
departures arise chiefly in connection with the relation of the 
weight of the pipe to the defining dimensions D and t, also in the 
varying price per pound which may exist as between lap -riveted 
and butt- strap riveted pipe, also in the effects arising from irregular 
profile and also in the influence on pipe line cost, of the specials 
required to connect together sections of different diameters. 

For these various reasons, any indication regarding economic 
diameter as given by formula should be checked with reference to 
the influence which such factors may have on the final values of 
the two terms X and Y, and on their rate of change for a small 
change in the diameter of the line. 

Reference should also be made to further reasons which may 
exist for decreasing the diameter of pipe toward the lower end, and 
quite independent of the problem of economic design. These are 
found in the increased difficulty of rolling heavy plate to circular 
form with increase in the thickness ; and also in the increasing 
difficulty of riveting up and handling extra heavy plates in the field. 
It is true that the difficulty in rolling heavy plate increases with the 
decrease in diameter as well as with the increase in thickness ; 
nevertheless, the thickness factor is on the whole the more impor- 
tant, so that while the decrease in diameter will partly offset the 
gain due to decreased thickness, there will result, at least within 
limits, some balance in facility of fabrication for the reduced size of 
pipe. Again, while there is no absolute limit to the thickness of 



MATERIALS, CONSTRUCTION, DESIGN 219 

plates which can be worked and assembled in the field, yet it is a 
well-known fact that the practical difficulties rapidly increase from 
1 inch or 1 J inches upward. Broadly speaking, plates 1 J to If inches 
in thickness are close to the limit of effective and satisfactory work 
in the field. If therefore the circumferential joints are to be riveted 
rather than flange joint connected, it is desiiable to hold the thick- 
ness not to sensibly exceed 1 J inches. But the thickness with any 
given pressure is directly proportional to the diameter. Hence by a 
suitable reduction in diameter at the lower end the thickness may 
be brought within an acceptable limit. 

Thus by way of illustration. Given a total static head of 1400 f. 
Then p=-433x 1400=606. Suppose that the use of equation (14) 
in this case indicates for the lower end of the line a diameter of 
60 inches with a thickness of If inches, and suppose that this 
thickness is considered undesirable and that it should be reduced to 
ly F inches. This will correspond to a diameter of 52 inches. These 
values will not be economic, but may be considered necessary, due 
to the reasons noted above. The question will then arise as to what 
extent this departure from economic conditions may be compen- 
sated for in the remainder or in the upper parts of the line. 

A first approximation to such compensation may be arrived at as 
follows : 

The economic lower diameter is 60 inches with If inches thick- 
ness. Suppose the minimum thickness employed at upper end to be 
■fc inch. Then the range of thickness is in the ratio of 5 to 22. 
Equation (14) shows that the economic diameters vary as the sixth 
root of the thickness. Hence the diameter range will be (4-4)*= 1-28 
and if the lower economic diameter is 60 inches the upper would be 
76-8. If then instead of an economic line with diameter tapering 
from 60 inches to 76-8, we substitute a line with taper from 52 to 
say 84 inches, thus giving approximately the same mean diameter, 
there will be an approach toward compensation and such a dis- 
tribution will give at least a first approximation toward the most 
economic line under the limitations imposed — a distribution of 
diameters which may be checked by one or two trial layouts on 
either side of that proposed. 

Determination of Thickness. — The thickness of the metal forming 
the shell of the pipe must be adequate to meet the following 
requirements : 

(1) The maximum static or steady condition pressure to which 
the pipe is likely to be subjected in operation. 

(2) Any excess pressure due to shock or water hammer to which 
the pipe may be exposed. 

(3) Collapsing pressure due to a reduction of the pressure on the 
inside to less than the atmospheric pressure. 

(4) Stiffness and rigidity and strength to meet such bending 
stresses as may be set up due to the action of the pipe as a beam. 

(5) Any reasonable special stresses to which the pipe may be 



220 HYDRAULICS OF PIPE LINES 

subjected as a result of expansion and contraction under tempera- 
ture changes. 

(6) Special stress due to distortion of the cross section of pipe 
from true circular form, or failure to realize this form as in lap- 
riveted pipe. 

(7) The usual variations to be expected in the strength of the 
plates resulting from the accidents of manufacture. 

(8) Corrosion and wear, in order that the pipe may have a life- 
time of reasonable length and without a too rapid decline of the 
factor of safety. 

In order that requirements (3), (4), (8) may properly be met, a 
lower limit is usually placed on the thickness, regardless of the 
strength called for by the other requirements. This lower limit, as 
elsewhere noted, is usually not far from one -quarter inch. At some 
point in the line as C, Fig. 123, the thickness required to insure the 
necessary strength against internal pressure will be £ inch. Then 
from this point to the upper end of the line the thickness is held 
uniform at this value, while from this point down it is increased 
in accordance with the strength requirements of the case. It must 
not be assumed, however, that a thickness of J inch will insure 
against collapse under external excess pressure. This will depend 
on the diameter. This thickness will, however, with the usual 
sizes, give a fair collapsing strength under a slight excess of external 
pressure and the air relief valve, as noted in Sec. 74, must be 
depended on to prevent any greater external excess than can be 
safely carried by this thickness. 

Requirements (1) and (2) relate directly to stress resulting from 
internal pressure. In order to meet these requirements with due 
allowance for the others, as in all engineering work, it is customary 
to base the design on the load which can be determined with some 
degree of certainty and to include all other unknown loads and all 
margin of uncertainty in the so-called factor of safety. In the 
present case we may design primarily with reference to require- 
ment (1) and depend on the factor of safety to provide sufficient 
margin for all of the other requirements and for the general out- 
standing margin of assurance desired ; or, otherwise, we may add to 
the pressure representing requirement (1) a certain amount, often 
taken from 50 to 100 (pi2) and representing requirement (2). The 
factor of safety need then include only the remaining items with the 
outstanding margin of assurance. 

With modern approved appliances for controlling excess pressure 
due to shock (see Sec. 75), the amount to be anticipated should not 
be too great to permit of merging with the other uncertain quantities 
under a fairly liberal factor of safety. 

The preceding remark regarding requirement (3) should not be 
here forgotten. The factor of safety is not expected to necessarily 
insure against collapse. It may or may not, depending on the 
thickness used. |In a pipe under high head it will undoubtedly be 



MATERIALS, CONSTRUCTION, DESIGN 221 

thick enough at the lower end to withstand full collapsing pressure, 
but may not be toward the upper end where the thickness ap- 
proaches the lower limit assigned for the line. 

As a still further or somewhat different factor of safety, a constant 
thickness is sometimes added, as in Fanning's formula for cast-iron 
pipe (see Sec. 58), in order especially to provide for numbers (7), (8). 

Assuming that all requirements other than (1) are to be covered 
by the factor of safety, it becomes of importance to inquire what 
values may be given to this factor. Experience indicates that 
values from (4) to (5) will insure a safe and satisfactory design. 
Furthermore, since the longitudinal joint is the weakest part of a 
riveted or welded pipe under internal pressure, the use of such 
factors of safety implies actual working stresses in the longitudinal 
joint of from 12,000 to 15,000 or 16,000 (pi2), assuming normal soft 
steel plates with an ultimate tensile strength of about 60,000 (pi2). 
With a joint efficiency of e this will imply an actual working stress 
in the plate itself, along a longitudinal line, reduced in the ratio =e/l. 



u y 

Fig. 124. — Layout of Thickness on Pressure Head. 

Passing now to actual methods of design we note first the 
formula from mechanics : 

pD=2teT (16) 

where p= pressure (pi2) 
D= diameter (i) 
£=thickness (i) 

e=emciency of longitudinal joint 
* T=safe working stress in metal of joint (pi2). 
Transforming into terms of thickness, we have 

HS < 17 > 

But p=wH, where 17=head at the given point and w=«433 = 
factor for transforming head of water in feet into pressure in pi2 
Substituting we have 

t- wHD (18) 

It thus appears with fixed values of e and T, that the thickness 
will vary directly as the product HD ; or with constant diameter, it 
will vary directly as the head H. 

In selecting the actual thickness to be used, convenient use may 
be made of the diagram of Fig. 124. First assuming the diameter 
uniform, lay off the line OY to represent the head (H+H ) (Fig. 123). 



222 



HYDRAULICS OF PIPE LINES 



OB is then laid off to represent the value of t at the lower end, 
assuming an actual joint stress 12,000 to 16,000 (pi2) as may be 
decided upon. The line BY, using OF as a base, will then give the 
corresponding value of t at any elevation. At C this line runs into 
the minimum thickness line AC, which is laid off at the desired 
distance from Y. The actual values for any given elevation will 
then lie along ACB. The indications of such a diagram may usually 
be very closely realized by commercial sizes varying by 16ths of an 
inch. Such a diagram set up on the sheet with a profile of the line 
will indicate at a glance the lengths and locations corresponding to 
the various thicknesses to be used. 

In case the diameter varies, increasing from bottom to top, the 
variation will not be continuous but stepwise. The diagram 
corresponding to Fig. 124 for such a case will therefore show a 
series of steps as in Fig. 125, where three different values of D are 
indicated. 

To solve equation (18) for quick approximate values, a form of 




Fig. 125. — Layout op Thickness on Pressure Head. 



straight -line diagram may be conveniently employed as illustrated 
in Fig. 126. In this diagram AB and CD are parallel lines with 00 
a perpendicular between them. The values of t in steps of J- inch, 
for example, are laid off on OB with any convenient unit. The scale 
for e is laid off similarly on OD, and for D on OC. Then OA becomes 
the axis of H and the unit of the scale will be given by the relation : 

TT . . , „ (Unit of t) X (Unit of e) X -433 

<p Unit of H =- ' \ — ^ 

(unit of D) x 2T 

Thus if 1 inch of thickness be represented by 3-2 inches on the axis 
OB, 100% efficiency by 5 inches on OD, 100 inches diameter by 
5 inches on OC, and putting T= 16,000, then unit of £=3-2, unit of 
e=6, unit of D=-05 and unit of #=-00433 or 100 feet=-433 inch, 
and the scale may be laid down accordingly. 

The diagram then fulfils the condition that any two lines drawn, 
as shown, and meeting on the axis 00 will fulfil the conditions of 
(18) and thus either one of H, t, e or D may be readily found once 
the other three are known. 

In this general connection Tables XXVI and XXVII will be of 
value, the former giving the weight of lap-riveted pipes and the 
latter of double butt-strap triple -riveted pipes, each for various 
diameters and thicknesses of plate as noted. 

These weights are based respectively on 6-foot courses and 
8 -foot courses, each with the usual allowances for overweight of 



MATERIALS, CONSTRUCTION, DESIGN 



223 




224 



HYDRAULICS OF PIPE LINES 



rolled plates, and include the calculated weights of the joints and 
rivets and a coating of asphalt.* 



TABLE XXVI 



Diameter 
(inside) 

inches 1/8 



18 
20 
22 
24 



33-9 
37-4 
411 

44-5 



Weights, in Pounds per Foot, of Lap-Riveted Pipes 
Thickness of Plates (inch) . 
3/16 1/4 5/16 3/8 7/16 1/2 9/16 



49-5 
54-6 
59-7 
64-4 



64-7 
710 

77-6 
83-8 



78-6 

86-5 

94-3 

102-0 



93-6 
102-7 
1121 
1211 



109-9 
120-5 
1311 
141-7 



126-7 
138-6 
150-9 
162-8 



1470 
160-7 
174-4 

188-0 



27 
30 
33 
36 



49-9 
55-3 
60-8 
66-0 



72-2 
79-8 
87-6 
951 



93-7 
103-3 
1131 
122-3 



113-9 
125-3 
137-2 
148-7 



1351 
148-7 
162-7 
176-5 



157-8 
173-5 
189-7 
205-4 



1810 
1991 
217-4 
235-4 



208-4 
228-8 
249-4 
269-9 



39 
42 

48 
54 



71-3 

76-6 
87-3 
98-0 



102-5 
110-2 
125-3 
140-6 



131-6 
141-8 
161-0 
180-2 



160-3 
172-2 
195-5 
218-9 



190-4 
204-0 
231-8 
259-2 



221-5 
237-4 
260-3 
301 1 



253-6 
271-5 
307-9 
3441 



290-2 
310-7 
351-9 
392-8 



60 
66 

72 



108-8 
119-4 
1300 



155-8 
170-8 
186-1 



199-5 
218-7 

237-8 



242-5 
265-6 

288-8 



286-9 
314-5 
342-0 



3330 
364-9 
396-7 



380-4 
416-6 

452-8 



433-9 
474-6 
515-6 



TABLE XXVII 



Weight, in Pounds per Foot, of Butt-Strap Riveted Pipes. 



Diameter 
(inside) 




Thickness of Plates (inch) 






inches 


i 1/2 


9/16 


5/8 


11/16 


3/4 


13/16 


7/8 


[15/16 


1 


24 


205-9 


2301 


255-7 


279-6 


3071 


350-3 


384-8 


426-5 


450-3 


27 


225-8 


253-2 


280-0 


306-7 


336-8 


384-2 


4191 


464-7 


490-3 


30 


245-6 


275-3 


304-7 


3331 


365-5 


416-6 


4550 


502-4 


531-0 


33 


265-7 


297-5 


329-3 


359-7 


394-2 


4500 


489-7 


540-3 


571 : 8 


36 


286-2 


320-7 


353-8 


386-4 


423-5 


482-9 


525-3 


579-4 


612-5 


39 


305-5 


342-3 


378-8 


413-8 


453-4 


515-3 


571-7 


618-5 


652-8 


42 


3260 


3650 


403-6 


441-0 


4821 


547-6 


595-0 


655-8 


6930 


48 


367-5 


411-3 


454-2 


495-7 


543-7 


616-7 


667-9 


734-3 


777-5 


54 


407-4 


455-7 


503-6 


549-4 


602-6 


681-8 


738-7 


8110 


857-9 


60 


446-8 


500-9 


553-8 


605-0 


6610 


748-4 


809-0 


886-6 


940-3 


66 


487-6 


545-3 


604 


657-6 


720-6 


814-2 


879-9 


964-5 


1020-7 


72 


527-9 


589-4 


652-6 


711-8 


780-0 


881-0 


951-4 


1039-5 


11021 



* See paper by Mr. J. D. Galloway, "Trans. Am. Soc. E.C., 1915," Vol. 
LXXIX. 



MATERIALS, CONSTRUCTION, DESIGN 225 

Thickness of Plates (inch) 

1-1/16 1-1/8 1-3/16 1-1/4 1-5/16 1-3/8 1-7/16 1-1/2 

27 524-4 5660 

30 5670 611-9 6531 686-4 

33 609-9 657-8 702-5 736-6 7761 809-2 

36 6541 704-9 751-3 788-2 831-8 864-9 908-8 949-9 

39 690-3 750-9 799-2 838-1 884-1 920-4 964-8 1009-9 

42 739-3 796-3 846-5 889-4 9361 976-4 1025-2 1071-3 

48 826-6 891-2 946-9 994-5 1046-9 1091-6 1144-7 1195-9 

54 912-0 983-7 1044-7 1096-8 1153-5 1204-4 1260-9 1318-5 

60 1004-9 10761 1142-2.1199-6 1261-2 1318-2 1380-9 1443-7 

66 1086-5 11691 1239-3 1301-7 1369-5 14271 1494-8 1561-7 

72 1171-6 1260-6 13361 1401-7 1473-5 1539-9 1612-9 1682-4 



68. Expansion and Contraction in Pipe Lines due 
to Changes of Temperature 

The well-known formulae of physics give us the following : 

L 2 =L 1 (l±at) 
Where L 2 and L x are the two lengths, t is the change in tempera- 
ture and a is the coefficient of linear expansion. 

Taking t in Fahrenheit degrees we have for a as follows : 

Cast Iron .... -0000056 

Wt. Iron and Steel . . . -0000064 

Brass -0000100 

Copper -0000089 

It is often more convenient to take the expansion for L ± =100 feet 
= 1200 inches and £=100°F. This gives numbers 120,000 times the 
above, as follows : 

Cast iron . . . . -66 (i) 

Wt. Iron and Steel . . . -768 (i) 

Brass 1-20 (i) 

Copper 1-07 (i) 

As a closely approximate value we may take for steel pipe an 
expansion of £ inch per 100 feet per 100 °F. 

In the case of an empty pipe, temperature changes of 100° are 
by no means impossible, especially in countries where the skies 
are clear and the sun bright. The total expansion of a line of 
pipe of considerable length is therefore a quantity requiring definite 
and careful consideration. Thus a line 1000 feet long will show a 
total expansion for 100° rise in temperature of approximately 
7-5 inches ; for a rise of 50° only, an expansion of 3-75 inches, and 
otherwise in proportion. 



226 HYDEAULICS OF PIPE LINES 

69. Stresses in Pipe Lines due to Expansion 
and Contraction 

Suppose a pipe line of length L rigidly held between abutments 
so that it cannot expand. Let the temperature rise t degrees, what 
stress will be developed ? We may treat this problem by assuming 
one of the abutments removed and the pipe free to expand, and 
then ask how much stress will be developed by compressing it at the 
higher temperature back to the original length. 

Denote the stress by T. Then by definition of the coefficient of 

elasticity E, we have 

AL L 2 -L t T 

— or — ~= — 

L L x E 

But hrZ^lJ^^ at 

Hence T=Eat. 
Inserting values for E and a for the four materials as above we 
have in each case an equation of the form, 

where B has values as follows : 

Cast Iron . . 56 to 112 (higher values for lower 
Wt. Iron and Steel . 192 values of t). 

Brass . . 100 

Copper . . . 140 

It thus appears, for example, that rise of temperature of 100 °F. 
in a steel pipe if held between abutments in such manner as to 
prevent expansion will result in a compressive stress of about 
19,000 (pi2). Similarly in the case of a fall of temperature in a pipe 
held between fixed anchors and prevented thereby from contracting. 
Precisely the same conditions will develop as above with the pipe 
under tension instead of compression, and the coefficients relating 
tension to change in temperature will be the same as above. 

In the case of riveted pipe with sections connected with circum- 
ferential riveted joints, the sectional area of the rivets in shear will 
normally be less than that of the pipe itself in tension or compression. 
Let m denote the ratio of the area of section of shell to the section 
of rivets. Then the stress which must be carried by the rivets 
(except as partly taken by the friction of the plates) will be m times 
as great as the T above. Since the value of m may vary from 1-3 to 
1*5 it is clear that under extreme conditions of temperature change 
stresses may develop which will result either in the rupture of the 
circumferential joints or in straining them beyond the elastic limit. 
Similar conditions may develop with flange joints under tension. 

The relief of pipe line from stress when subject to temperature 
change is realized through some form of slip or expansion joint (see 
Sees. 49, 77). 



MATERIALS, CONSTRUCTION, DESIGN 227 

70. Erection of Steel Pipe Lines 

The chief points involved in the erection of a steel pipe line are 
the following : 

(1) The supply of suitable provision for carrying the down -hill 
thrust of the line and for safeguarding the permanent connections 
at the lower end (power units, pump discharge, etc.) from undue 
stress or disturbance due to such thrust, before the line has finally 
settled into its permanent condition. 

(2) Safeguarding the line at all points against undue stress due to 
changes of temperature. 

(3) The proper support and constraint of the pipe at all points 
during erection and before all parts are completely assembled. 

(4) Hydrostatic test in sections of not too great length so that 
any weakness or faulty construction may be promptly discovered 
and remedied. 

Certain of these requirements are of special significance only in 
the case of pipes laid on steep slopes, as with power plant penstocks. 
Others are more general in character. In what immediately follows 
we shall have especially in view the case of steep -slope construction. 

The above various requirements in the case of steep -slope con- 
struction are usually best met by erection from the lower end 
upward. Requirement (1) should be met by the provision of an 
especially strong and reliable anchor block capable of safely carry- 
ing all the down -hill thrust that can in any way be anticipated. In 
addition, many good engineers prefer to leave out, between the 
lower end of the line and the permanent connections, a short flange - 
joint filler piece which is not put in until the line is installed and 
has had time to settle into permanent condition. This filler piece 
may then be cut to the proper length and inserted in place. In 
any case, however, the anchor block should be installed between 
the line and the permanent connections, as a means of absorbing 
all down -hill thrust and of shielding them from the stress which 
such thrust might produce. , 

Requirement (2) will be met without additional features in case 
the design contemplates an adequate supply of expansion joints. 
In such cases, however, it must be borne in mind that the pipe 
fixed at any given anchor block will push up the hill and draw back 
down the hill as the temperature rises and falls. Care should 
therefore be taken, and especially when the pipe is empty and in 
climates subject to wide fluctuations of temperature, to avoid any 
temporary constraint which might hamper or prevent such tempera- 
ture movements. 

In case the pipe, when exposed to the sun, can be filled with 
water, the temperature movements are greatly reduced, usually to 
a negligible amount. It is therefore always desirable to keep the 
pipe from the lower end upward filled with water as the erection 



228 HYDRAULICS OF PIPE LINES 

proceeds, both with reference to the reduction of temperature 
movements and for purposes of strength and leakage tests as 
referred to below. 

gAgain, in cases where the pipe is to be buried or where for other 
reason expansion joints are omitted, the same care must be 
exercised during installation and while the pipe is uncovered, to 
keep the line filled so far as possible and to avoid hampering or 
preventing such movements as accidental changes in temperature 
may produce. 

Again, if for any reason a line without expansion joints is placed 
under complete constraint at two distant points and while still 
uncovered, special care must be taken to safeguard the pipe against 
undue temperature stress. Filling with water will usually meet 
every requirement. If this is not practicable, temporary shelter 
from the sun may be necessary. These various special precautions 
regarding temperature movements are, of course, the less necessary 
as the line between points of constraint is curved or provided with 
angles or bends. 

Requirement (3) calls only for the exercise of normal engineering 
judgment and care. The tendency toward movement under 
gravity or any other forces which may be involved, must be care- 
fully evaluated, and if the features of the permanent design are in 
any degree inadequate during the process of erection they must be 
supplemented with suitable means according to the requirements 
of the case. 

Requirement (4) calls for periodic hydrostatic tests. To this end 
there are required the following : 

1. Means for closing the section of line to be tested. 

2. Means for applying the test pressure desired. 

The lower end of the line may usually be closed without difficulty. 
If it is connected through to permanent units, such as a waterwheel 
or pump, there will be an intervening valve, the closure of which 
will meet all requirements at this point. If it is not connected 
through to permanent units, the lower open end will usually be 
provided with a flange. In such case a test blank flange should be 
provided, by means of which the lower end may be temporarily 
closed for test purposes and also for holding water in the line as 
installed, for reasons as noted above. 

In case there is an open plain end without flange, closure may be 
effected by means of a temporary bulkhead (see Sec. 78). 



71. Piers and Anchors 

Pipe lines require definite points of support in order that the 
weight when filled with water may be carried without danger of 
serious deformation through sagging. Likewise at critical points, 
especially where changes in direction are made, and at intermediate 



MATERIALS, CONSTRUCTION, DESIGN 



229 



points on long straight grades (especially when steep), definite points 
of complete constraint or anchorage are required. 

Forms and Construction of Piers. — In modern engineering 
practice piers are commonly made of concrete, a 1 : 2 : 4 or 1 : 2 : 5 
mixture of cement, sand and crushed rock representing standard 
proportions. Where the pier is intended to serve simply as a 
support for the weight of the pipe it is usually made of sufficient 
width to embrace about one-third or sometimes nearly one-half 
of the circumference. 

Where several lines are carried along parallel and closely adjacent, 
the piers will naturally combine into a series of continuous concrete 
walls underlying the pipes at suitable intervals and formed with 
properly rounded depressions to receive and constrain the pipe. 

Due to expansion and contraction resulting from changes in 
temperature, the pipes must slide back and forth on the supporting 
surface of the pier. This slipping movement will be greatly facili- 
tated with relief of stress, both in the pipe and in the pier, by the 




Fig. 127. — Saddles for Support of Pipe Line. 



use of some form of metal bearing surface. A convenient and 
effective form of bearer for this purpose is shown in Fig. 127. 
AB is a cast-iron ring formed to the curvature of the pipe and with 
a rib C on the convex or lower side. This rib is received in a groove 
formed in the concrete pier, and the ring AB is thus kept in place 
during the travel of the pipe back and forth. The friction of steel 
pipe on cast iron is much less than on concrete, and the pipe will 
therefore slip with comparatively small resistance on the surface of 
support. Furthermore, the slipping of a pipe back and forth 
directly on the concrete surface will tend to break off and crumble 
the pier at the edges DE, and in case an iron ring support is not used, 
the edges of the piers at D and E should be well bevelled away in 
order to save them from such action of the pipe. 

In the setting of piers in pipe -line construction, the point of 
primary importance is that of sub -foundation. The pier is intended 
to furnish a secure and dependable point of support for the pipe 
line. This cannot be realized unless the pier itself has also a secure 
and dependable foundation, either on bed rock or on hard pan or 
other stratum not liable to yield or shift or to suffer damage from 



230 HYDKAULICS OF PIPE LINES 

local wash. In the best practice the piers should be anchored into 
the sub -foundation (preferably bed rock) by vertical steel rods. 
Old railroad rails, or material otherwise possibly scrap, may often 
be utilized to excellent advantage in this manner. This anchoring 
of the piers into the sub-foundation is of special importance on 
relatively steep grades. Instead of anchoring the pier into the bed 
rock by steel rods, as above suggested, it will sometimes be prefer- 
able to excavate into the rock and thus anchor the pier by pouring 
the concrete into the cavity, formed preferably with rough and 
overhanging sides. The entire question of the sub -foundation for 
piers and of the extent to which they should be tied or anchored 
into such foundation is one for the exercise of good engineering 
judgment, having in view the purpose of the pier and the local 
conditions applying to the case. 

Anchors. — In addition to definite points of support, pipe lines 
require, at critical points such as angles and Y branches and at 
certain intervals on long straight grades, definite points of attach- 
ment or constraint. 

These are intended to furnish fixed points in the line for the 
anchorage of expansion joints and relative to which expansion and 
contraction can take place. Likewise they are intended to supply, 
especially on long uniform gradients, a suitable number of points 
of complete constraint relative to movement of the pipe in all 
directions. 

In approved practice anchor blocks are made of concrete of the 
same mixture as for piers, and preferably with sufficient steel 
reinforce to insure strength and coherence under any stresses' 
which can be anticipated. The question of sub -foundation is, of 
course, of special importance, and wherever practicable the anchor 
block should go down to bed rock or to an entirely dependable 
formation. Regarding anchorage to or into the sub -foundation, 
the same remarks apply as in the case of piers. Here, however, 
such anchorage should be considered as absolutely essential. In 
no other way can safety be assured against longitudinal movement 
under the stresses to which the line is subject. 

Special means must also be provided for securing the pipe to the 
block in order that the latter may fulfil its purpose. Such modes of 
attachment are usually of two kinds. 

(a) A circumferential strap passing over the upper part of the 
pipe and secured by bolts or equivalent fastenings anchored into 
the block. 

(b) A projecting rib or flange, formed by an angle or T bar riveted 
around the pipe and bedded in the block. 

Very commonly the concrete of the anchor block is carried up 
and over so as to entirely surround the pipe. A strap as in (a) acts 
primarily to prevent lateral displacement. When properly designed 
and set up, however, it may also be depended on to give security 
against longitudinal movement. A ring or flange as in (b) acts 



MATEKIALS, CONSTRUCTION, DESIGN 231 

primarily to prevent longitudinal movement. If, however, the 
concrete block with suitable steel reinforce is carried over the pipe, 
the latter will operate as a band or tie, thus providing security 
against movement in both directions. 

In the design of anchor blocks, as with piers, much will depend on 
the local conditions and special circumstances of the case. The 
engineer must simply hold in mind the ultimate purpose of the 
structure, and then adopt such means as the conditions may 
indicate. 



72. Relative Advantages of Buried or 
Unburied Pipe 

In many cases, such, for example, as municipal water supply, 
convenience will require that pipe lines be buried. In other cases, 
as, for example, a water-power penstock line on a rock hill side, the 
expense of burying may be practically prohibitive. In such cases 
the matter may be determined by necessity and independent of 
questions of relative advantage or disadvantage otherwise. In 
many cases, however, the conditions will be such as to admit of 
either mode of treatment as may be judged most advantageous. 
The various points of advantage and disadvantage may be summar- 
ized as follows : 

Regarding all matters related to inspection, repairs, upkeep 
generally and length of serviceable life, the advantages will lie with 
the unburied pipe. Such lines may be readily inspected as to leaks, 
condition of protective covering and condition generally. Repairs 
such as the calking up of leaky seams, are readily carried out. 
Repainting and general maintenance are also carried out under 
convenient conditions. Naturally all of these conditions, if taken 
advantage of, will make for the maximum serviceable life of the 
line. On the other hand, the buried line cannot be inspected or 
repaired or repainted on the outside, and these conditions will 
naturally reduce the serviceable life of the line. 

With regard to stresses developed under changing temperatures 
and the necessity of making provision for them by slip joints or 
otherwise, the advantage lies with the buried pipe. As noted in 
Sec. 70 with the line continuously full of water, such stresses are 
commonly reduced to a negligible amount. But pipe lines must be 
unwatered from time to time for internal inspection or repair and 
under these conditions, if exposed to the sun, changes in length may 
develop with severe stress, unless such changes are adequately 
accommodated by slip joints or suitably located bends. With 
buried pipe the line is protected from the direct action of the sun 
and the variation in temperature of the pipe, even if not carrying 
water, will be very much reduced. Under these conditions the 
stresses developed under changes of air temperature are usually 



232 HYDKAULICS OF PIPE LINES 

negligible, and the design will not therefore require the provision 
of special expansion joints, as in the case of unburied pipe. 

With regard to cost of laying there is no certain advantage with 
either side. The cost of ditching and backfilling will be saved in the 
case of the unburied pipe. Even here, however, a certain amount 
of ditching will be required in order to eliminate small irregularities 
in the ground. On the other hand, the cost of pier supports and 
anchors will normally be less for buried than for unburied pipe. 
Piers and anchors cannot in all cases be dispensed with in the case 
of buried lines. Careful judgment will be required in accordance 
with the circumstances of the case. If the trench is carried through 
a hard permanent formation, the requirement of support, as such, 
will be adequately met. Anchors for steep side hill work will, 
however, be required at points to be selected with judgment. If the 
trench is carried through soft uncertain formations, then both piers 
and anchors will be needed, distributed and spaced according to 
judgment. 



73. Protective Pipe Coatings 

The problem of the protective covering of steel or iron pipes is 
fundamentally the same as for all iron and steel structures. Whether 
open or buried, corrosive agencies begin immediately to attack 
unprotected surfaces and operate unceasingly to reduce the metallic 
wall of the pipe to the condition of crumbling oxides and metal 
salts. 

Such destructive agencies attack both the internal and external 
surfaces of the pipe, the former according to the possible chemical 
reactions between the metal of the pipe and the liquid carried and 
the latter according to the soil or earth formation in which the pipe 
is laid, alternations of moisture and dry-out, etc. 

The conditions vary somewhat according to whether the pipe is of 
cast iron or plate steel and whether buried or above ground. 

Cast-iron pipes are usually buried. There is no fundamental 
reason for this except that under the conditions specially suited to 
cast iron as regards pressure and type of service, convenience 
usually requires them to be placed underground. There is there- 
fore no opportunity for periodic examination and repainting or 
retreatment. Whatever is done must be done when the line is 
installed. 

Fortunately cast iron is relatively resistant to the attack of 
corrosive agencies and under normal conditions a life of twenty 
years or upwards may be anticipated. 

Any comprehensive treatment of the problem of pipe -line 
corrosion and its prevention is beyondfthe purpose of the present 
work, and we can only note briefly the fundamental conditions to 
be observed in carrying out such measures. 



MATERIALS, CONSTRUCTION, DESIGN 233 

The best protective coatings fall into two general classes — 
chemically neutral carbon or graphite paints and coatings of a 
bitumastic or asphaltic character. 

Where paints are applied the surface should be thoroughly 
cleaned of all scale, dirt, oil, or grease. A coat of red lead is then 
commonly applied as a base and the carbon or graphite paint over 
this. Especial care should be taken to secure the maximum of 
surface dryness as the paint is going on. Paint will not effectively 
adhere to a moist or wet surface. The careful cleaning of the surface 
is most important and the expenditure of a sum for cleaning, even 
approximating that of the paint itself, will be fully justified by the 
longer life of a coat of paint carefully laid on to a thoroughly cleaned 
and dry surface. 

In the case of bitumastic or asphaltic coverings, the most favour- 
able conditions are determined by a thoroughly cleaned surface, 
warm or hot liquid and a warm or hot dry surface. In the case of 
small pipe (cast-iron commercial pipe, etc.) the hot dip process is 
commonly employed, effectively realizing the temperature conditions 
indicated above. In the case of large pipe the covering must be 
applied by brush ; but even here, hot Uquid and a warm pipe 
surface are important aids in securing a closely adhering and 
effective covering. The best of such coatings when properly applied 
give a hard, enamel-like, closely adherent surface which will, for a 
long period of time, resist ordinary corrosive action. 
§5 All steel pipe lines, especially if buried, should be protected in the 
most effective possible way when installed. Especial care should 
be taken to see that the protective coating, whatever it may be, 
covers the pipe completely, that it is allowed to become dry and 
hard before backfilling and that it does not become abraded during 
the process of filling the trench. 

Steel pipe, if uncovered, should be likewise carefully painted or 
treated. In particular, care should be exercised to see that the pipe 
where it lies in the saddles is well covered, as also the saddles them- 
selves. The sliding back and forth due to temperature changes will 
inevitably abrade any coating which can be applied, but if the 
surfaces are liberally covered with protective material, the condition 
will be better than if left bare. 

Regarding the dipping process, it may be noted that cast iron is 
better adapted to this method than is steel. The looser texture and 
rough surface of cast iron seems better adapted to furnish a bond 
with the dip than is the relatively smooth surface of steel plate. On 
this account many engineers of experience prefer, for steel plate pipe, 
a paint coating, properly laid on as noted previously, as furnishing 
on the whole the best protection against corrosive agencies. $ 

Where pipe is laid in concrete and where it is desired that the 
concrete and iron shall bond together, as in an anchor block, the iron 
surface should be left clean and unpainted in order toTpermit 
bonding with the concrete. 



234 HYDKAULICS OF PIPE LINES 



74. Air Relief Valves 

In the operation of a pipe line, conditions may arise which, at 
certain points, may drop the hydraulic grade line below the level 
of the pipe ; see Sec. 14. This means that the pressure within the 
pipe will be reduced below the atmosphere and danger of collapse 
of the pipe may result. To prevent the development of such a 
condition or to control the subpressure within safe limits, an air 
relief valve may be fitted. This is, in effect, a form of safety valve 
opening inward and admitting air inside the pipe, thus preventing 
the pressure from dropping below an assumed safe limit. In the 
examination of this problem there are two main questions. 

1. What is the maximum drop in pressure which may be con- 
sidered safe in the case of a pipe with given diameter and 
thickness ? 

2. What aggregate area of air valves will be required in order to 
prevent the drop in pressure exceeding this safe limit. 

The first of these involves simply the question of the strength of 
the pipe under an excess external load ; the second involves the 
hydraulic characteristics of the line and the problem of the flow of 
air through the valve opening. 

Regarding the strength of large cylindrical pipe against collapse, 
there is great uncertainty so far as direct experimental evidence 
goes. Existing formulae give the most widely divergent results. 
It is well established, however, that the double thickness at the 
joints — lap or butt-strap — gives an added element of strength, and 
that a uniform pipe without such local stiffening rings would 
collapse under a lower pressure than actual pipe with such local 
stiffening furnished by the joint doubling. Taking advantage of 
this fact, relatively thin pipe is sometimes stiffened further against 
collapse by a riveted circumferential angle iron, in the middle of 
each length, thus furnishing a definite stiffening ring at these points. 

Among the various formulae proposed, the following by Love 
may be taken as giving an indication of the collapsing pressure for 
long uniform tubes or pipe. 

^=65,000,000 (tjDf (19) 

2)= pressure (pi2). 
2=thickness (i). 
D= diameter (i). 

This formula does not take account of any support derived from 
the doubling at the joints, and hence actual pipe is likely to show 
strength greater than as indicated by the formula. The error, 
therefore, is likely to be on the side of safety. 

If Love's formula is taken as primarily applicable when L/D—6 
or more, and if the collapsing strength for lesser values of LjD is 
taken as varying inversely as the square root of the length (all other 



MATERIALS, CONSTRUCTION, DESIGN 



235 



things the same), then an indication of the collapsing strength of 
pipe with angle -iron stiffening may be found as follows : 



Pl =2>5p\/ l 



Where p is the strength as derived from (19), D is diameter and L 
is length between angle -iron supports, both in the same units of 
measure. 

Thus for illustration for t=-25 and D=50 we shall find p=8 (pi2). 

If, however, we have angle -iron stiffening spaced at 100 -inch 
intervals we shall have \/D/L=-7 and p 1 =l'75p=14-0 (pi2). 

Strength against collapse is also affected adversely by any 
departure from a circular cross section. For this reason, other 
things equal, a pipe with lap -riveted longitudinal joint may collapse 
at lower pressure than if of true circular section. 

There is much need of further experimental data on the collapsing 

TV 




Fig. 128. — Design of Air Relief Valves. 



strength of large riveted steel pipes under external load, and no 
present formula can be depended on to do more than give some 
indication of the probable load which such a pipe may safely bear. 

The second part of the problem, involving the hydraulic char- 
acteristics of the line, together with the assumed emergency 
condition, includes a range of possibly variable quantities and 
conditions so wide as to make a general solution scarcely practicable. 
Typical cases will, however, serve to indicate the way in which an 
approach to a treatment of the problem may be made. 

In Fig. 128 let ABC denote a pipe line considered in two sections, 
L v L 2) as indicated. For simplicity we may take the diameter 
uniform throughout the length from A to C. The necessary 
modification for variable diameter will be readily made by reference 
to Sec. 8. 

Then from Chapter I (45) for steady flow conditions and with no 
air-valve opening we shall have, for the velocity in the line : 



v=\J i 



H 



+ - 



(20) 



236 HYDRAULICS OF PIPE LINES 

For the resultant pressure head at B we shall have 



$-*>-$+£)> < 21 > 



Now suppose that this value of v is one which gives a hydraulic 
grade line NDJ with pressure head at B, as in equation (21), 
measured by BD below the atmosphere. 

Next suppose that it is desired to raise the pressure head at B up 
to D l9 giving for AB a hydraulic gradient ND X instead of ND. 
This new pressure will then tend to reduce the velocity in AB below, 
and to raise that in BC above the original velocity v. 
g Let q denote the new pressure in the pipe at B and v x and v 2 the 
new velocities in AB and BC. 

We shall then have at B under the new conditions in L x 

vT 2g * C 2 r 

/H^qjw 
Whence v 1 ^\J 1 jj x (22) 

2g + C*r 
Likewise at C under the new conditions in BC we shall have, 
just outside the nozzle or opening, 

u * = V H V 2 2 , g , V 
20/ 2gfm 2 2 C 2 r~ r w~ r 2g 

[H 2 +q\w+v x 2 \2q 

Whence v a =V _1 L±_ (23) 

2gfm 2+ C 2 r 

In this equation no account is taken of any loss of head resulting 
from the more or less abrupt change of velocity from v x to v 2 . 

With q=p we should find v 1 =v 2 . With q>p we shall have 
v 2 >v 1 and the difference in velocity must be made up by the inflow 
of something at B. If the something were water, the problem would 
be simple. The amount of such inflow would be measured by 
(v 2 — v-l) multiplied by A, the c.s. area of pipe, and it would only 
remain to provide suitable means for securing the inflow of water 
at this rate. In the actual case, however, air is the substance the 
inflow of which is depended upon to maintain the pressure con- 
ditions desired, and such inflow is to be realized as a result of the 
difference between the pressure of the atmosphere and that within 
the pipe at the point B. The air thus drawn into the pipe is sus- 
ceptible to volume changes in accordance with the pressure changes 
between the atmosphere and within the pipe at B, and thence down 
the line to the point of exit at C. The presence of this variable 
element in the contents of the pipe greatly complicates the problem 
of pipe flow and renders analytical treatment exceedingly difficult, 
at least without some further experimental results bearing on the 
phenomena of such mixed flow. 



MATEKIALS, CONSTRUCTION, DESIGN 237 

% 

In the absence of any ready basis for precise treatment, guidance 
may be usually obtained by neglecting the change in the volume of 
the air along the pipe between B and (7, by assuming the mixed flow 
to be given by the same formulse and coefficients as for water alone, 
and in order to cover the divergence between such assumptions and a 
more exact hypothesis, by allowing a generous factor of safety. 

If then a is the aggregate area through the air valves and z the 
inflow velocity, we shall have 

za=(v 2 — v-JA 

or a= v - ? ^— (24) 

B z 

or wJgr!£P (25) 

z 
Where Ud 2 is the sum of the squares of the diameters of the 
valves and D is the diameter of the line. 

The velocity z as a function of the difference in pressure between 
the atmosphere and within the pipe at B, is given by Table XXVIII 
in which, however, the coefficient of inflow or efficiency of the valve 
considered as an orifice, is taken at 0-60. We have thus at hand 
through equations (22), (23), Table XXVIII and (25) all require- 
ments for a solution of the problem. 

A numerical example will illustrate the procedure. 
Given H 1= = 115(f). 
H 2 = 285 (f). 
L x = 800 (f). 
£ 2 =1000 (f). 
D= 4 (f). 
Thickness of plate at B= -25 (i), 

Then from equation (19) it appears that such a pipe would be in 
danger of collapse under a subpressure of about 9 pounds per 
square inch. 

Let it be proposed to maintain the subpressure at 4 pounds per 
square inch as a safe margin. Then from Table XXVIII, with an 
inflow coefficient of -60 and at sea-level, the velocity of inflow under 
this head=569 (fs). 

Suppose that a rupture of the pipe at C or other emergency 
condition gives a discharge opening equivalent to half the area of 
the pipe and that the discharge coefficient / for the opening may be 
taken at 0*80. Assume also the Chezy coefficient (7=110. 
Then in (20) we shall have 

#=400 
^+^2=1800 
/=-80 
m=-50 
(7=110 
r=l 
Substituting and reducing we find 
v=42-02. 



238 



HYDRAULICS OF PIPE LINES 



TABLE XXVIII 

Depression (pounds per square inch) 



Altitude 
feet 


2 


4 


6 


8 





392 


569 


718 


856 


1000 


399 


582 


735 


878 


2000 


407 


594 


752 


900 


3000 


416 


607 


770 


922 


4000 


424 


620 


788 


945 


5000 


433 


634 


806 


970 


6000 


442 


648 


825 


995 


7000 


451 


662 


845 


1023 


8000 


460 


677 


865 


1050 


9000 


470 


692 


887 


1079 


10000 


480 


708 


909 


1107 



Velocity of inflow of air through air valve (fs). 
Coefficient of discharge /is assumed =-60, 

and from (21) for the resultant pressure head we find 



If then we put q- 



^=-29-2/= -12-66 (pi2). 

-4 (pi2) or q/w= -9-23 (f ) we shall find from (22) 
^=39-00 



and from (23) t; a =43-21. 

Hence v 2 — ^=4-21. 

Then from Table XXVIII z=569. 

Hence we have from (25) 2d 2 — -^ — = 



17-05. 



This would indicate a single valve about 4-5 inches diameter or 
two 3 -inch valves. 

If this assumed condition were the most serious to be anticipated, 
then the provision of air values as indicated should be adequate. 
On the other hand, it is always possible that the entire lower end of 
the pipe might rupture out in such a way as to give an outlet opening 
equivalent to the entire c.s. area of the pipe. In this case we have 
m=l, and assuming/==l we should find in the above case v=49-34 
and with this velocity, for the pressure head at B, 

^=-83-8 (f). 
w 

If, again, the pressure is to be maintained at —4 (pi2) and the 

pressure head at —9-23 (f ) we shall have for v x the same condition 

and value as before, viz. ^=39-00, while from (23) we shall find 

v 2 =55-23. 

Hence v 2 —v 1 — 16-23 

, _., 16-23x2304 _ Ki70 

and 2d % — ^77 =65-72. 

5o9 



MATEKIALS, CONSTRUCTION, DESIGN 239 

This would indicate, say, two 6 -inch or three 5 -inch valves. 

Suppose, again, we assume a condition as in Fig. 129 where AB is 
a long line, BCD a so-called " inverted siphon " and DE a relatively 
short section. 

Suppose complete rupture at C as the most serious possible case. 
The condition of reduced pressure will develop quickly at B, but the 
velocity v t through L x will be slow in developing due to the long 
length and time lag. We may then take as an extreme case an 
assumed velocity in L x of v , the original steady motion value, and 
for v 2 in BG the value resulting from equation (23) assuming the 
pressure at B maintained at the desired limit value q below the 




Fig. 129. — Design of Air Relief Valves. 

atmosphere. We shall then have the basis for a determination of 
v 2 —v 1 and for Zd 2 , the aggregate square of the diameters for the 
air valves at B. 

Thus for numerical values let us take 

2^=21600 

L 2 =800 

L 3 =600 

1^=1000 

#!=100 

# 2 =200 
#3=160 
# 4 =300 
/at #=-90 
raat.#=-0771 
g/w;at£=-9-23 
C(Chezycoef.)=110 
D=2 (f). 
Then the overall length=24000 (f) and the overall head=440 (f) 
and from (20) we shall find for the steady motion velocity before 
rupture, v =S (fs). 

Then, after rupture at O, we apply equation (23) to the conditions 
in L 2 putting v 1 ~v Q =S and / and m—l. We thus find : 

v 2 =36-02. 
Then v 2 — ^=28-02 

and ^ = 28-02x576 =28 . 36 , 
ony 



240 HYDKAULICS OF PIPE LINES 

This indicates at B a single valve about 5*5 in diameter or two 
4 -inch valves. 

Again, at D we shall have water flowing in both directions — 
down DE to the discharge end and down DC to the point of 
upture. In such case we shall have in equation (25) the numerical 
sum of the two velocities instead of their difference. Suppose 
again the value of q at D to be maintained at— 4 (pi2). 

Then adapting equation (23) to the conditions in L 3 and the 
discharge at C, assuming/ and ra=l we shall have in the formulae, 
160 for H 2 , 600 for L 2y zero for v x and —9-23 for qjw. Whence we 
find ^=36-24. 

Again adapting the same equation (23) to the conditions in L A 
and the discharge at E we shall have in the formula, 300 for H 2 , 
1000 for L 2 , zero for v ly -9-23 for q/w, -9 for / and -0771 for m. We 
then find v 4 =9-73. 

The incoming air must supply the volume corresponding to both 

of these velocities. 

Hence we have v 3 -fv 4 = 45-97 

, _, a 45-97x576 . . _ 

and 2d 2 — —^ =46-54 

569 

This implies one 7-inch valve or two 5-inch valves. 



75. Pressure Relief Valves and Breaking Plates 

These items of pipe -line equipment are intended to operate as 
safeguards against the development of an undue excess pressure as 
a result of sudden valve movement under the various conditions 
discussed in Chapter III. Pressure -relief valves fall under two 
general classes according as they are intended to operate auto- 
matically, consequent upon a slight or limit rise in pressure, or as 
they are attached to some part of the control mechanism of a power 
unit thus deriving their movement from the movement of the latter. 
Valves of the latter type fall rather under the category of hydraulic 
power plant equipment and therefore lie beyond the scope of the 
present work. 

Valves of the former type may be considered as an item of pipe- 
line equipment and as such merit brief notice. 

While there are many types and structural forms of such valves, 
their operation depends on substantially the same basic principles. 
Fig. 130 shows in diagrammatic form the characteristics of one of 
the best of such automatic valves. 

The valve itself is seated under a slight excess of pressure due to a 
difference in area between the valve and the balance piston. The 
main pipe line is connected through a small pipe to the space under 
the operating piston as shown. In this connection, and not shown 
in the drawing, is a valve under the control of a small pilot valve 
held normally closed under a balance of pressure between the main 



MATERIALS, CONSTRUCTION, DESIGN 



241 



pipe line and the expansion tank. On the arrival of a pressure wave 
along the pipe line this balance is disturbed, the valve in the pipe 
connection is opened and the excess pressure builds up under the 
operating piston, resulting in the opening of the main valve and the 
relief of the pressure. 

On the return toward normal pressure the pressure in the 
expansion tank will determine the return of the pilot valve, the 
closure of the connection through the small pipe and the resultant 
closure of the main valve due to the over balance of area. Any 
desired degree of retardation in the movement of the main valve is 
secured through the adjustable by -pass connecting the two ends of 
the operating cylinder. 

All such valves depend for their operation on an initial rise in 
pressure, and the assumption of effective operation depends on the 



e.OJUSTftat-E. 



E.XPANSIOM TANK 
COMPLETE-LY FILLED W<TH WATER 




Ol6CMAR.GiS PI»M 



R8.Lie.ff VflLVt 

Fig. 130. — Automatic Pressure Relief Valve. * 



possibility of an adequate response to the initial pressure rise within 
the time permitted by the character of the pressure -time history. 
References to Chapter III will show the conditions under which the 
pressure rise may be expected to be abrupt in time or gradual. If 
the initial rise in pressure is sufficiently slow the operating parts of 
the valve may have abundant time in which to perform, each its 
function, and thus the effective operation of the whole may be 
secured. If, on the other hand, the case is one of very abrupt 
pressure rise, as in the case of the closure of a pipe line running with 
valve opening nearly or quite the full size of pipe and hence nearly 
on " gravity flow " (see Chapter III, Fig. 48), the rise of pressure 
just at the instant of valve closure is exceedingly rapid, constituting 
in effect a hammer blow, and no valve could be expected tor operate 
in such time as to safeguard the line from such blow. 

* Under Patents of the Pelton Water Wheel Co. 



h.p.l. — R 



242 HYDKAULICS OF PIPE LINES 

Too much dependence should not therefore be placed on such 
automatic valves, and in particular their time characteristics of 
operation should be closely studied in connection with the probable 
time histories of the pressure shocks which are to be anticipated, and 
assurance should be obtained that within the probable period of 
pressure rise to the desired limit value there will be time for the 
effective operation of the valve in relief of pressures beyond such 
limit. 

Breaking Plates. — As a further safeguard in connection with the 
rapid rise of pressure due to water ram, the use of the breaking plate 
should not be overlooked. This is a plate of appropriate area, 
bolted on at the lower or delivery end of the line, somewhat as a 
manhole cover-plate, but designed and intended to rupture under 
an excess pressure well below that which the pipe itself may be 
expected safely to bear. The area of the opening covered by such 
plate may be preferably some two or three times the nozzle or 
normal discharge area, thus insuring an immediate relief of pressure 
consequent on the rupture of the plate. 

A first approximation to the design of such a plate may be made 
through equation (26) of Chapter IV, using for p the over pressure 
under which it is desired that the plate should break, and for the 
denominator, values some four times those given for safe operation. 
It is, however, always desirable to have such design checked up by 
actual test under the limit pressure desired, and in order to eliminate 
the uncertainties of all empirical formulae and the unknown influ- 
ence due to slight variation in the physical properties of the 
materials employed. 

76. Manholes and Covers 

In order to permit of access to the interior of large pipe lines for 
examination, painting, re-caulking, etc., manholes with cover plates 
are provided at occasional points in the line. The metal around 
such a hole is reinforced by a suitable doubling plate with inner 
steel casting for cover-plate joint. The cover is fitted up and 
secured in place making joint against the inner surface of the 
reinforce casting, entirely in accordance with the practice f amiliar in 
steam boilers and which need not be here described in detail. 



77. Expansion Joints 

The purpose of expansion joints has already been discussed in 
Sees. 69, 70. In the best practice a given section of the fine which is 
to be treated as a unit in the matter of expansion and contraction is 
stabilized at its lower or delivery end by a suitable anchor block and 
provided at its upper end with an expansion joint. The pipe then 
creeps up and down the slope or back and forth along the line from 
the anchor block to the joint. 



MATERIALS, CONSTRUCTION, DESIGN 243 

The characteristic or essential elements of an expansion joint are 
indicated in Figs. 96, 97. The mode of operation and general 
character of construction aside from full structural details will be 
evident from the figures. As noted in Sec. 50, such joints are some- 
times provided with guard bolts to prevent complete separation of 
the two parts of the joint in case of an extreme temperature drop. 

In all cases the special hydraulic forces which develop as a result 
of the introduction of an expansion joint must be carefully examined 
and care taken to provide the necessary support or constraint to the 
two parts of the joint in order to prevent separation under the 
operation of such forces. 



78. Test Flanges and Test Bulkheads 

In connection with a program of test on large pipe lines it may 
become necessary to close temporarily an open end of the line. For 
this purpose a special so-called " test flange " is employed. This 
consists in effect of a cover plate, usually of cast steel, suitably 
ribbed or reinforced to safely bear the anticipated load, and 
provided with flange at the rim for connection to the corresponding 
flange on the open end of the pipe. If there is no such flange 
connection on the pipe it may become necessary to provide a 
corresponding " companion flange " and rivet it to the open end in 
order to realize the closure. 

The general manner of dealing with the problem of structural 
design in the case of such a test flange has been outlined in Sec. 53. 
■ It may also become necessary, in connection with the same 
program of test, to segregate out a special part or section of the 
line for individual test. To this end one or possibly two test bulk- 
heads may be employed. A test bulkhead is some form of structure 
built up in such fashion as to permit of location within the pipe at 
the desired point and provided with means for deriving from the 
shell of the pipe the necessary support under the anticipated load. 
In the case of light pressures such a structure may be built up of 
timber and made tight by oakum calking or other like means. If 
the line is made up of in and out sections, support for the bulkhead 
may usually be derived from the end of the pipe section of smaller 
diameter. If of wood, the bulkhead at the rim would, in such case, 
need a metal plate reinforce in order to safely carry the load against 
the end of the inner pipe without crushing. If the pressures are 
heavier, some form of steel bulkhead will be preferable, fitted with 
an adjustable packing ring for making the joint against the shell 
and with hinged feet which may swing out and bear against the end 
of the smaller pipe or against a group of rivet heads, thus carrying 
the load. With the test concluded these feet may then be swung in, 
the joint unpacked and the bulkhead removed or shifted along to 
the next location as desired. 



244 HYDRAULICS OF PIPE LINES 

In cases where the pipe is of uniform diameter on the inside, 
thus offering no convenient end for the support of the load, but 
where the pressures are not too extreme, the bulkhead may be 
supported against cross timbers secured by wedging friction against 
the sides of the shell. The principle involved in this mode of 
support is illustrated in Fig. 131. AB is a heavy timber carried at 
iona plate of steel as a local reinforce to the shell of the pipe. 
At B it rests on a plank cut slightly wedging. Under these con- 
ditions pressure against A would promptly dislodge the timber, 
while, on the other hand, pressure at B would develop the wedging 
action giving an end thrust at A and corresponding reaction at B, 
with consequent friction grip on the shell of the pipe. The end B 
of such a timber is therefore a point capable of carrying a heavy 
thrust in the direction of the arrow. Four or six such timbers 




spaced around the circle of the pipe will then give a series of points 
such as B, and from which, by suitable blocking or thrust timbers, 
the support may be carried back to the bulkhead itself. 

79. Pipe-Line Valves 

The general subject of pipe -line valves is too extended to permit 
of adequate discussion in the present work. It is indeed a subject to 
which an entire volume might well be devoted. No attempt will be 
made therefore to discuss the subject here, either from the descrip- 
tive or the design standpoints. 

Space may be permitted, however, for a few words relating to the 
importance of valve design and to the main problems which present 
themselves in this connection. 

The importance of a design which shall insure safe and reliable 
operation over a long period of time cannot be well exaggerated. 
There has, in many cases, been a tendency to design valves for large 
pipe -line work. — valves up to 4 or 5 feet in diameter — simply as an 
overgrowth of ordinary small pipe valves and without adequate 
recognition of the severe conditions of service under which such 
large hydraulic valves are intended to operate. 

A large valve of this character must meet adequately the follow- 
ing major requirements. 



MATERIALS, CONSTRUCTION, DESIGN 245 

1. The bocfy or casing with its supporting ribs must be of suffi- 
cient thickness and character of design to insure, under the highest 
over-pressures to be anticipated, not only safety against rupture 
but also against any sensible deformation of any of the parts, such 
as might determine either leakage or binding and jamming of 
moving parts. 

2. Strength of moving parts and of operating gear adequate to 
close the valve under full flow of water. 

3. Bearing areas sufficient to insure operation of all sliding or 
rubbing surfaces over long periods of time and under the highest 
pressures anticipated, without danger of seizure or abrasion. The 
proper selection and use of metals for the two parts of a mutually 
sliding pair (as a valve gate and its seat) will aid in marked degree in 
the realization of this requirement. 

Pipe -line valves are made in three general forms : 

1. Gate or slide valves. 

2. Butterfly valves. 

3. Bulb or nozzle valves. 

These forms will all be familiar to those in contact with hydraulic 
work, and with individual interpretation the above conditions may 
be applied to all of these forms or types of valve. 

In particular it may be recommended to accept designs predicated 
upon the actual conditions to be met and carried out by those 
familiar with such work rather than so-called stock designs developed 
to meet, as a stock article, a certain average range of operative 
conditions, but frequently lacking in features which may have 
special importance for the particular problem in hand. 



80. Pipe-Line Fittings, Ys, Bends, etc. 

In accordance with the express purpose of the present work, as 
noted at the head of this chapter, no attempt will be made to 
discuss, in a descriptive manner, the various forms of pipe-line 
fittings. Brief reference has been made in Chapter IV to certain of 
the problems of structural design which may arise in connection 
with such forms. The material commonly employed in all large 
high-pressure work is cast steel, reinforced and ribbed as the 
characteristics of the case may require. 

In the case of all such large fittings, as in the case of valves, and 
especially under high pressures, use may be recommended of 
special designs developed by those familiar with such work rather 
than of any form of so-called stock or standard design developed 
without special reference to the peculiar conditions of the case in 
hand. 



CHAPTER VI 

OIL PIPE LINES 

81. Physical Properties of Oil Affecting Pipe 
Line Flow 

Due to the greatly increased value of the viscosity of oils, as com- 
pared with that of water, the problem of the flow of oil in pipe lines 
presents certain special features which will be briefly considered in 
the present chapter. 

The general problem of the flow of viscous liquids in pipe -lines is 
included within that of the general problem of pipe line flow as 
discussed in Appendix I. 

For many pumping problems it may be convenient to transform 
the expression for pressure gradient, as in Appendix I, (4), into 
pounds per square inch per mile of pipe line. As defined, the 
pressure gradient G is in pounds per square foot per foot of line. 
We have therefore to divide by 144 and multiply by 5280. We may 
also reduce D to inches by introducing the factor 12, thus giving 

finall y n 440/cri; 2 

° m= ~D~ 2g * (1) 

Where 6r, n =gradient in (pi2) per mile of line. 
/= coefficient as in Appendix I. 
g= density (pf3). 
D= diameter (i). 
v= velocity in (fs). 

For any given case it becomes, therefore, necessary to determine 
for the conditions of operation, the values of /x and a, the viscosity 
and density, and thence to proceed as indicated. 

The density of oils is usually indicated by the so-called gravity on 
the Baume scale, taken at some standard temperature, usually 
60° F. 

In order, therefore, to determine the value of the argument 
Dva/fA, the following information is required : 

(1) The diameter D. 

(2) The velocity of flow v. 

(3) The relation between gravity in Baume degrees and density a. 

(4) The relation between density at standard temperature of 60° 
and density at any specified working temperature. 

246 



OIL PIPE LINES 247 

(5) The value of the viscosity (jl at the specified temperature, or 
if this is not known directly, such relations between viscosity, 
gravity and temperature as will make possible some estimate of the 
viscosity. 

With this information at hand both jx and a may be determined 
or estimated and thence the value of the abscissae Bva\\i. Thence 
with due regard for the roughness of the pipe, we may assume a 
value of /and thence from (1) determine either the friction head or 
the pressure gradient per mile. 

Relation (3) is provided by the formula 

„_ 8736 . 

where B= gravity on Baume scale 

and a = density in pounds per cubic foot. 
Relation (4) is provided by the formula 

y=yt-^=^<t-m°).. : . (3) 

where y 1 = density in pounds per cubic foot at 60° F. 
and y— density at temperature t. 

This is an empirical relation which has been found to agree 
closely with observations on American petroleum oils. 

Relation (5) cannot be definitely developed simply because 
observation shows that the viscosity of oils is not determined by 
gravity and temperature alone. Viscosity apparently depends on 
the number and proportion of the various complex constituents of 
which the oil is composed, and on their physical states, as well as 
on the overall resultant gravity and temperature. 

Broadly it is found that viscosity increases with increasing 
density or decreasing gravity Baume, and with decreasing tempera- 
ture. The relation of viscosity to gravity is, however, irregular in 
special cases and subject to occasional exception. 

With regard to the variation with temperature it is usually found, 
starting with a high temperature, that the increase of viscosity is at 
first slow, the rate of increase rising rapidly as the temperature 
drops. 

Professor W. R. Eckart of Stanford University has pointed out 
that the relation between viscosity and temperature t when plotted 
on double logarithmic scales shows a close approximation to a 
straight line, at least over the working range of temperatures for 
which the liquid may be said to retain its identity. At high 
temperatures, the more volatile constituents will begin to vaporize, 
and at very low temperatures certain constituents may begin to 
solidify and separate^out, thus in either case changing the character 
of the liquid itself. Between these limits of temperature, however, 
Professor Eckart has shown^byfa large number of cases that within 
the limits of observational error the straight -line relation may, for 
all practical purposes, be assumed to hold. 



248 



HYDEAULICS OF PIPE LINES 



It thus results that if the value of /x for a given oil is known for 
two temperatures, the logarithmic plot may be drawn as a straight 
line between these points and extended over the working range of 
temperatures, thus giving a direct and practical form of relation 
between viscosity and temperature, and specifically, the value of /x 
for any specified temperature within the working range. 

Likewise in the relation between density and viscosity at a fixed 
temperature, there is evidence of a slow rate of increase of viscosity 
with density at low values of the density, followed by increasing rate 
with increasing values of the density. 

For numerical values we have the following : 

A. C. McLaughlin* gives diagrams showing the relation between 
temperature F. and absolute viscosity for a series of American oils 
from which the following values may be drawn : 







TABLE XXIX 




Temperatures 


Fah. 




Viscosi 


*t 


60° 


. 




. -060 to 


•300. 


80° 










•040 to 


•250. 


100° 










•020 to 


•160. 


120° 










•016 to 


•080. 


140° 










•013 to 


•045. 



The densities of the oils to which these values refer range from 
56-8 (pf3) to 60 (pf3). (Gravity Baume 24° to 16° app.) 

From tests made by Cooper on 60 samples of California petroleums 
R. P. McLaughlin! gives values for the relation between viscosity 
and gravity from which the following tabular values are derived : 



aume Lbs. per cub. ft. v 
ravity Density at 60° Fah. 


iscosity at 60° Fah.t V 


iscosity at 185° 


36° . 


. 52-6 . 


•0040 . 


. -00076 


34° . 


. 53-3 . 


•0050 . 


. 00090 


32° . 


53-9 . 


•0070 . 


. -00120 


30° . 


54-6 . 


•0103 . 


•00160 


28° . 


. 55-3 . 


•0145 . 


•00206 


26° . 


56-0 . 


•0206 . 


•00260 


24° . 


. 56-7 . 


•0280 . 


•00320 


22° . 


57-5 . 


•0370 . 


•00380 


20° . 


58-2 . 


•0760 . 


•00440 


18° . 


59-0 . 


•2640 . 


•00650 


16° . 


59-8 . 


— 


•01120 


14° . 


60-7 . 


— 


•02450 


12° . 


61-5 . 


— 


•14800 



* "Journal Am. Soc. Mech. Eng., 1915," p. 263. 

f The units involved in these values are the poundal, foot, second, 

j "Journal, Am. Soc. Mech. Eng., 1915," p. 264. 



OIL PIPE LINES 249 

The very rapid increase of viscosity with density will be noted 
for 60° beginning about density =57 and for 185° beginning about 
density =60. 

From a number of tests made on California oils Dyer* gives for 
oils of four different gravities values of the viscosity for varying 
temperatures from which the following tabular values are derived. 







TABLE XXXI 






Gravity Baume 

Density, lbs. per 

cub. ft. at 60° 


18°-2 
58-95 


18°-6 
59-59 i 


15° 

60-26 


12°-1 
61-48 


Temp. Fah. 

50 


•7360 


Viscosity t/* 






60 


— 


1 -0530 


— 


— 


75 


•2450 


•4970 1 


•5670 


— 


100 


•1220 


•2230 


•3440 


— 


110 


— 


— 


— 


1-6610 


125 


•0670 


•0990 


•1500 


•6140 


150 


•0365 


•0490 


•0750 


•2110 


175 


•0242 


•0245 


•0310 


•1020 


200 


•0216 


•0220 


•0220 


•0510 



The above oils are noted to have contained about 2% of 
water. 

The numerical measure of viscosity is often met with in terms of 
accepted forms of viscosimeters such as the Engler, Say bolt or 
Redwood, or again it is not infrequently stated in terms of the 
metric units, centimeter, dyne, second, or again in terms of the 
more recently proposed unit the " centipoise." 

For the latter we have the following definition : 

One absolute metric unit (c.g.s.)=100 centipoises. 

Hence to convert viscosity in absolute units into centipoise units, 
multiply by 100. To convert viscosity in centipoises into absolute 
units, divide by 100. 

To convert absolute metric units into absolute English units, or 
vice versa, we have the following relation : Viscosity (English units) 
= Viscosity (metric units)-^ 14-88 (poundal, foot, second) (dyne, 
centimeter, second). 

Regarding the various forms of viscosimeters, the following 

* "Journal, Am. Soc. Mech. Eng., 1915," p. 259. 

f The units involved in these values are the poundal, foot, second. 



250 HYDRAULICS OF PIPE LINES 

formula will serve for transforming the indications of these instru- 
ments into absolute English units : 

^=•00000237^-^1^ (Saybolt)* ) 
<j t 



^=•00000158*-^^ (Engler) 
o t 

^=•00000280^-^15? (Redwood) 
a t 



<*) 



Where /x,= viscosity in absolute English units (poundal, foot, 
second). 
cr=density in pounds per cubic foot. 
£=time on instrument (Saybolt, Engler, Redwood) 
(seconds). 
In illustration of the use of the equations of the present section, 
suppose we have given as follows : 

Size of pipe . . .8 (i). 

Capacity per day . . 24,000 bbls. 

Gravity of oil . . 18° B. 

Average temperature . 100° Fah. 

Viscosity (assumed) . *12. 

We find firsts =4-47 (fs). 

Then from (2) density at 60° F.- =59-02 
and from (3) density at 100° F. =58-10. 
We then find the value of D v o\\i = 1 443 . 

This implies stream line motion and (8) of Appendix I gives a 
value of / about -0444. 

Substituting this in (1) we find the pressure gradient 44-1 pounds 
per square inch per mile of length. 

Again, if we should take the oil of gravity 16° B. at 60° F. with a 
working temperature of about 110° F. and a value of /x=-15 we 
should find similarly a value of ZW/ju,=1166, a value of /about -055 
and a pressure gradient 6r m =55-2 pounds per square inch per mile 
of length. 

Again, suppose a 10-inch line with oil of gravity 20° B. at 60° F. 
and at an average working temperature of 110° F. and with a 
velocity of 5 feet per second. We may take jit about -06. We then 
find as above for the density at 110° F., cr= 57-06. 

This gives a value of ZW/ju,=3964, implying turbulent flow and 
for commercially smooth pipe a value of / about -04 (see Table, Ap- 
pendix I). If 10 per cent is added for a slightly roughened surface we 

* The numerical constants in the Saybolt equation refer to the recently- 
standardized form and proportions of the instrument. The indications of 
earlier instruments will not quite agree among themselves or with any one 
set of values. The values for the Redwood instrument are somewhat less well 
established than for the other two, Bee U.S. Bureau of Standards Technologic 
Papers, Nos. 100, 112. 



OIL PIPE LINES 251 

have /=-044 and substitution in (1) gives a pressure gradient of 
42-95 pounds per square inch per mile of length. 

Suppose again that we have a record of £=1000 for the Saybolt 
viscosity of a given oil at 1C0° F. Then from (4) we have /x/ct= 
•002368 and <r//x=422-3. Again, if £=4000 we shall have /x/cr= 
•00948 and a /fi= 105-5. If in the first case the oil is one showing a 
Baume gravity at 60° of 17-5 we shall have from (2) a at 60°=59-22 
and from (3) a at 100°=58-31 and hence ju at 100°=58-31 X -002368 
=•1381. 

From an examination of the form of the curve ABCD, Appendix I, 
it is clear that the conditions of operation should, if possible, be so 
chosen so as to avoid a value of the abscissa Dvcr/fi at or just 
beyond the critical value. In the case of oil pipe lines the value 
will often fall close about this point. Obviously if the conditions 
admit of control they should be so adjusted as to give a value on the 
stream line branch AB, and as near the critical velocity as practic- 
able without actually passing the limit. This will insure the lowest 
practicable value of the friction head coefficient /. Decrease in the 
abscissa value will mean a rapidly rising value of / on the branch 
AB, while a slight increase will mean rapid rise of the value to the 
branch CD. 

82. Oil Pipe Lines 

In the sense here employed the term oil pipe line is intended to 
refer to a line for the transportation of crude or fuel petroleum oil 
in bulk from the wells to convenient rail or water shipping points or 
to refinery locations. The principles involved in the discussion of 
pipe -fine resistance are of course entirely general. The descriptive 
matter and the suggestions for design, however, are intended to 
more directly apply to the case of large and long pipe lines as above 
noted. Such pipe lines are usually of steel pipe of diameters from 
6 to 12 inches. In the United States 8 inches is a common size. 
Such pipe is made of thickness suitable to stand a test pressure of 
1200 (pi2) and with a safe working pressure of 800 (pi2). The pumps 
for handling the oil at the pumping stations are quite commonly 
made to meet the same pressure requirements, the number of such 
stations being determined by the capacity and size of the line, the 
length and the topographical characteristics. 

The influence of viscosity on oil pipe line resistance has been 
noted in the preceding section, also the dependence of viscosity on 
temperature. To reduce the viscosity, especially with heavy oils, 
heaters are commonly employed, one at each station and often one 
intermediate between stations. 

The station heaters are commonly formed of closed steel cylinders 
provided internally with headers and tubes through which the oil 
usually makes two passes on its way from the receiving tank to the 
pumps. The exhaust steam from the pumps passes between the 
headers and tubes and raises the oil to an initial temperature 
ranging usually from 125° to 150° F. 



252 HYDKAULICS OF PIPE LINES 

The heaters used between stations consist of a by-pass manifold 
of 4-inch pipes lying at right angles to the main line, often 200 or 
250 feet long, and with headers so arranged that the oil passes out 
and back a distance of 400 to 500 feet through the 4-inch pipes. 
These pipes are carried in a brick chamber or flue along which 
passes the hot gas from a furnace at one end, burning oil drawn 
from the line through a suitable reducing valve. With the pressure 
suitably reduced an atomizing burner of the usual type may be 
employed, giving with suitable air control a nearly smokeless com- 
bustion, the gasses from which pass along the flue as above noted. 
By these means the temperature of the oil may be raised some 30 
to 50° F. at these midway points. 

The pumps commonly used are of two types : (1) direct-acting 
duplex plunger with tandem compound or triple -expansion 
steam cylinders, and (2) the crank and flywheel, plunger fitted 
pumping engine with cross compound or triple -expansion steam 
cylinders fitted with Corliss valves. The design throughout must be 
exceptionally strong and rugged and the proportions such that with 
a steam pressure of 135 (pi2) at the boilers an effective pumping 
pressure of 800 to 1000 (pi2) may be realized. The plungers are 
usually from 6 to 9 inches diameter by about 36 inches stroke, those 
for direct acting pumps ranging larger than those for the flywheel 
type. Such pumps will handle about 1000 bbls. of oil per hour, the 
flywheel type at a somewhat higher number of strokes per minute. 

The economy of the flywheel type is naturally superior to that of 
the direct acting, but the latter are found necessary for starting the 
oil in a long line after a shut-down. The steady direct thrust which 
can be realized by the direct -acting pump is found much better 
suited to overcome the resistance of a long line of cold oil in starting 
from a condition of rest, than the effort derived from a pump of the 
crank and flywheel type. In this connection it may be noted that a 
condition of rest is always avoided so far as possible, especially with 
cold weather or with heavy oil. In fact, the combination of the two 
may render starting up from a state of rest extremely difficult if not 
impossible. It is therefore a principle of operation that once the 
column of oil is in motion it must be kept moving at all hazards, so 
far as it is humanly possible to realize such end. 

The construction of an oil pipe fine offers but few points calling for 
special comment. The line is usually placed underground to a 
greater or less depth depending on the temperature conditions along 
the line. The joints are of the screw-collar type, usually with 
special design so far as length is concerned in order to insure a 
tight joint under the high pressures employed. 

Protective coverings or coatings are employed, consisting usually 
of bitulithic enamels or hot asphaltum followed with a wrapping of 
roofing paper, which is again coated with asphaltum. Heat- 
insulating coverings to reduce the loss of heat from the oil would 
be desirable, but cannot usually be justified economically. 



OIL PIPE LINES 253 

The line is commonly tested by pumping water through under 
pressure from the station pumps, the line being carefully inspected 
in the meantime for leaks at the joints or flaws in the pipe. 

Expansion joints are used to some extent, though not as much as 
good design would seem to indicate. Without such means for 
accommodating the variations due to changing temperatures, the 
stresses developed must be absorbed in the line itself ; and, as 
noted in Sec. 69, this may cause serious stresses resulting in leaky 
joints and trouble. 

The serious effect of low temperatures and consequent increased 
viscosity on pipe-line resistance, especially with heavy oils, is 
shown by the changing capacity of pipe lines dependent on the 
operating conditions. Thus an 8 -inch line in California is stated to 
have had its capacity of 25,000 barrels per day with medium oil in 
summer reduced to about 3600 per day with heavy oil in winter. 

The capacity of an 8-inch line is usually taken from 20,000 to 
30,000 bbls. per day when working under conditions not excessively 
severe regarding temperature or gravity. Similarly for a 6-inch line, 
which is usually worked at a pressure somewhat higher than the 
8-inch, the capacity is commonly taken from 12,000 to 18,000 bbls. 
per day. 

83. Design of General Characteristics of Line 

The factors which enter into the determination of the general 
characteristics of an oil pipe line are the following : 

(1) The diameter of the line. 

(2) The length of the line. 

(3) The physical properties of the oil to be handled. 

(4) The capacity of the line. 

(5) The pressure to be realized by the pumps. 

(6) The topography and profile along the proposed location of 

the line. 

(7) The distribution of the pumping stations. 

Very commonly the first six of these are known or assumed and 
it is required to find the last, the most suitable distribution of the 
pumping stations. 

We shall outline briefly the steps whereby this problem may be 
readily investigated. 

Conditions (1), (3), (4) will serve to determine the frictional 
resistance and hence the pressure gradient due to friction along the 
line from station to station. 

If the physical conditions of the oil remained uniform throughout 
the run from one station to the next, the gradient would be a 
constant and the hydraulic grade line would be straight and 
inclined at the constant gradient value. Due, however, to the 
falling temperature, the viscosity and with it the friction coefficient 



254 HYDRAULICS OF PIPE LINES 

/, will show constantly changing values, and hence a gradient 
changing with progress along the line. The hydraulic grade line is 
therefore no longer straight, but curved. 

If data are at hand permitting an assumption of the probable 
temperature distance history, that is of the probable average 
temperature for each mile of run, the corresponding gradients may 
be determined and thus for any given distance the resulting 
hydraulic grade line determined. 

In some cases the so-called logarithmic decrement law for 
relating temperature change to distance has been used. This is in 
the form AH?l\ 

\t-Tj x 



log 



log 



t -T 



(5) 



Where 2 =initial temperature of oil. 
t= temperature at distance x. 
t x — temperature at distance x v 
T= temperature of earth or air surrounding pipe. 
#=any distance along line. 

a; 1 =a known length of run for which t x is known or 
assumed. 

Hence if we know, in any given case, the length of run x, the 
final temperature at this point t v the initial temperature t and the 
temperature T, we may readily determine, on this hypothesis, a law 
for temperature change with distance, and determine the hydraulic 
grade line accordingly. 

So long as the conditions of flow are either continuously stream 
line or continuously turbulent (see diagram Appendix I), falling 
temperature and increasing viscosity will result in a continuously 
increasing value of / and a continuously steeper and steeper 
gradient. If the conditions should change during the run from 
turbulent to stream line flow (as may readily be the case) the values 
of / may show first an increase, followed by a sudden decrease in 
passing from one condition to the other, and then followed by a 
continuous increase for further cooling under stream line conditions. 
The actual form of the grade line will be, therefore, more or less 
complex, depending on the history of the conditions of flow along 
the Hne^ 

The use of such a grade line, once determined, will be best 
illustrated by an example. Assume, therefore, data as follows : 

1. Diameter of line : 8 (i). 

2. Length of line : 144 miles. 

3. Density of oil : Baume gravity 18. 

4. Capacity of line : 27,000 bbls. per day. 

5. Pressure of pumps : 800 (pi2). 



OIL PIPE LINES 



255 



In Fig. 132 let XX be a base line on which are laid off linear 
distances along the pipe line. Then at suitable intervals, and from 
the topographic data, altitudes above an arbitrary datum are 
erected. In this case we take station O as the datum and thus 
derive, as shown by the curve, a distance — altitude history OABC. 
. . . Note should be taken that this is not a profile in the more 
usual sense of the term. It is not a history of altitude on horizontal 
distance, but a history of altitude on linear distance along the line. 

We next find v=5-03, and then based on the best estimates which 
can be made regarding the temperature history with distance and 
regarding the variation of viscosity with temperature, we obtain by 
equation (1) values of the pressure gradient for successive miles of 
run. Suppose for illustration the first three of such gradients to be 
49-5, 50-0, 50-7 pounds per square inch. Then the total head used 









\ 




\ 

\ 








2 




2500 




\ 


\i 




\ 












2000 


\ 


\ 


\ 


r~~^ 


t \ 












fyOO 


\ 

A 
/K, 


\ K 2 




\ 


K \ 




F 


? P 


2000 






\ 










1000 


\ K 3 


K< 


\d 


500 




V 


X 





\ 














""\^/ 




















X 



Fig. 132. — Oil Pipe Line Design — Graphical Construction. 



will be 49-5 for one mile, 99-5 for two miles and 150-2 for three miles. 
In this manner we may prepare a continuous history of total pressure 
drop on distance. It will next be convenient to transform these 
pressure values into head of oil in feet. To this end we may use an 
average value of the density since the variation of density with 
temperature is relatively small. With such values we then plot (to 
the same scale as in Fig. 132) the hydraulic grade line on stiff paper 
and cut out as a template. Let PQR denote such a template 
covering a total pressure drop PQ of 1000 pounds, or about 2500 feet 
for the case in hand, and a total distance PR of 16 miles. The total 
head furnished by the pump is 800 (pi2), or about 2000 feet head 
of oil. We then place the template with head scale vertical and with 
the 2000 foot point on the head scale (counting downward from Q) at 
the home station 0. The grade line QR intersects the distance - 
altitude line in A and at a distance from measured by OK v At 
this point and for this distance it is clear that the height through 
which the oil has been lifted is measured by K X A, while the head 



256 HYDKAULICS OF PIPE LINES 

used in pumping the given distance is AL. The sum of these make 
up K X L the total head available from the pump. The point K x thus 
determined gives therefore the location of the next station beyond 
0. We next transfer the template to A as starting-point and repeat 
the operation, thus determining the successive points B, C, D, etc., 
and thence, the locations of the stations K v K 2 , K 3 , etc.* 

When we pass the crest of the hill and descend as from D or iT 4 on , 
we shall have a total head available made up of the head due to the 
pump plus the head due to the oil, and the head used in friction will 
then equal the sum of these two. The same procedure, however, will 
determine properly the location of the station as indicated for the 
run from D to E. 

In this manner we continue throughout the length of the line, 
finding in this case the last station at /, or not quite at the end of the 
line. This implies, of course, a slight readjustment of values, either 
a slightly higher pumping pressure or a slightly lower velocity or a 
higher general temperature for the oil. Some readjustment may 
also be required in order to avoid undesirable locations topographi- 
cally for the pumping stations. Such readjustments may involve 
extra heater provision, or the use of a larger pipe, or of a double line 
for a certain distance. These various problems of secondary adjust- 
ment do not, however, involve any new hydraulic principles, but call 
rather for the exercise of sound engineering judgment in the light 
of all the factors bearing on the problem in hand. 

If the effect of changing temperature is neglected and an average 
temperature assumed, leading to a single value of the pressure 
gradient, the grade line will become straight and the template of 
Fig. 132 will become a right-angled triangle with the grade line as 
the hypothenuse. The general program of use is naturally the same 
as for the template of Fig. 132. In the case of oil of low density 
(high Baume) no heating may be required, and the temperature 
will remain substantially uniform throughout the run. In such 
case the hydraulic gradient naturally becomes a straight line, with 
procedure as indicated above. 

There are many other points of detail which will arise in con- 
nection with studies of this character, and the present chapter is 
only to be considered as a brief sketch of the hydraulic principles 
'involved and of the general mode of treatment through which they 
may be effectively applied to the problem. 

* In the diagram of Fig. 132 the scale is naturally very much reduced. 
In dealing with an actual problem such scales should be employed as will 
insure the degree of accuracy significant for the case in hand. 



APPENDIX I 



GENERAL THEORY OF PIPE LINE FLOW 

The phenomena of pipe line flow will obviously depend on the following 
factors, or conditions defining the circumstances of the case : 

Diameter of pipe denoted by D 

Density of liquid „ a 

Velocity of flow „ v 

Viscosity of liquid „ /x 

Length of pipe „ L 

Character of pipe surface. 

From the first four of the above determining characteristics there 
will result : 

Pressure gradient required to overcome 
resistance to flow, or otherwise, the 
loss of pressure per unit length due to 
resistance to flow, denoted by G 

From G and L will result : 

Total loss of pressure head in line, denoted by h 

The theory of dimensions applied to the problem of pipe line flow 

shows that there must subsist between these quantities a relation of 

the form* 



*-Tr*(^0- (1 > 



D r \ p 

where 9 denotes some function of the quantity \Dv cr//x). 

This equation takes cognizance of all factors in the problem except 
the character of the pipe surface. We have, moreover, for character 
of surface or degree of roughness, no definition nor unit of measure, 
and hence the influence due to this factor must remain to be allowed 
for within the numerical values of the function «p(Z>i;o , //x). 

With G known we have then', for the total loss of pressure head in 
the line : 

hJ^ (2) 

cr 

In the . above equation the viscosity /x is the so-called absolute 
viscosity defined by the equation R—fiv/z, where R is the force opposing 
the motion of a plane of unit area lying very near and moving with 

* See among other references, Buckingham, "Trans. Am. Soc. Mech. 
Eng.," Vol. XXXVII, p. 263. 

h.p.l.— s 257 



258 



HYDRAULICS OF PIPE LINES 



a velocity v parallel to a large plane, the space of the thickness z between 
the two planes being filled with the liquid in question. 

If, then, in equation (1) the form of the function 9 could be deter- 
mined, this equation would give a complete solution of the problem 
of pipe line flow, at least so far as determined by the quantities repre- 
sented therein, and hence for any one value of roughness or condition 
of pipe surface. 

In order to assimilate this formulae to those more commonly employed 
in hydraulic problems we may take the Darcy formula (see Sec. 5), 



7>= 



fLv* 
D 2g 



(3) 



equating the values in (2) and (3) we have 

a J D2g 



or G= 



2gD' 



(4) 



Combining this with (1) we find 

/ (Dvo\ 



2<7 



/MVCT 



or 



/ =2<7<p ( 



Dvo\ 



(5) 



In this equation the value of the abscissa [Dva/fi) is independent of 
the system of units employed — English or Metric — so long as the 
system is homogeneous ; that is, a given quantity always in terms of 
1'4 A\ 

P3 
12 
M 
10 
09 
.08 
.07 
.06 
05 
04 
cgj.09 



01 

00 



1000 2000 



10 000 



100.000 



Log. Scale of Dvo/n 
Fio. 133. — Diagram op Values of/ on abscissa op Dfo-ffi 



APPENDIX I 259 

the same unit. We shall assume here, however, English units used 
throughout, feet, pounds and seconds. 

We have, therefore, in any specific case, to take the values of D, 
v, a and ft and find the value of (Dvcr/fi). If, then, we know or can 
estimate the value of the function for this value of the argument 
{DvG/fJb) we may multiply by 2g and thus find the coefficient / in the 
familiar Darcy formula. If desired we may then readily find the 
Chezy coefficient C from the relation between / and C as developed 
in Sec. 5. 

Now experiments with smooth brass and steel pipe made with 
such diverse substances as air, water and oil, agree in giving for the 
relation between / and (Dva/fx) a curve of the general form shown in 
Fig. 133. 

The part of the curve from A to B corresponds to the so-called pure 
stream line or irrotational flow. In this mode of flow there is no 
turbulence or eddy formation and the paths of the liquid particles are 
smooth, open, straight or gently curving stream lines. The part of 
the curve from C to D corresponds to turbulent or rotational flow. 
In this mode of flow the liquid is turbulent with eddy formation and the 
paths of the particles are curving, twisted and contorted, as may 
result from the accidents of the turbulent flow. 

Experience shows that the transition from one mode of flow to the 
other occurs abruptly at or near a so-called critical value of the abscissa 
(Dvcr/fx). It appears furthermore that this critical value is commonly 
found between 2000 and 2500 as indicated in the diagram. Further, 
as shown by the form of the curve, the transition from stream line to 
turbulent flow is accompanied by a sudden increase in the value of 
the ordinate /, followed later by a gradual decrease with further in- 
creasing values of the abscissa. 

It appears further that for stream line flow the pressure gradient G 
varies directly with the velocity. Reference to equation (1) shows 
that this requires a form of relation 



G=-— K\ ~— ) where K is a constant. 



That is, the form of the function 9 must be simply the reciprocal 
of the abscissa (Dvcr/fi). Hence from (5) we must have for this case 

^ K (£) < 6 > 

If, then, we denote the abscissa (Dvcr/fi) by #, we shall have 

xf=2gK (7) 

This equation shows, therefore, that for pure stream line flow the 
curve AB is a hyperbola determined as in (6). 

Again, experience shows for turbulent flow that the gradient G 
varies nearly with the square of the speed. The index for ordinary 
values of the abscissa and for ordinary degrees of roughness of pipe is 
usually found slightly less than 2, often close about 1-85. For very 
rough pipe, however, and as the abscissa increases, the index ap- 
proaches close to the value 2. It is instructive to note, so far as the 
value of the abscissa is concerned, that the approach to the index 2 
is nearer as D is greater, as v is greater, and a is greater and as fx is 
less. 



260 HYDRAULICS OF PIPE LINES 

At the limit, when we may assume G to vary as v 1 , we shall have 
from (1) 



9 Kir) 



constant. 

That is, the curve CD will reach and maintain itself, as a straight 
line parallel to the axis of X and giving an ordinate equal to / for this 
limiting condition. 

It therefore appears that the curve CD must be considered as 
gradually approaching the horizontal as a limiting condition, at which 
point we shall have /= constant and Goov*. 

In Table XXXII are given values of / for various values of the 
abscissa (Dva/fx). Between abscissa values 200 to 2000 inclusive the 
values of / correspond to the branch A B representing stream line flow, 
These are derived from the hyperbola 

(/) (£>i>o>)=2<7#=64. (See (7)) (8) 

Intermediate values are readily determined from the equation. 

From abscissa value 2500 upward the values correspond to the 
branch CD representing turbulent flow. 

These values of /, which are closely accurate for such widely diverse 
substances as air, water and oil, are derived from experiments on brass 
or smooth steel pipe. 

Regarding the influence of roughness on these values, it appears 
that for stream line flow the character of the surface of the pipe seems 
to have but slight influence. On the other hand, it does have a direct 
and important bearing on the phenomena of turbulent flow. It results 
that the values of /, or of ^(Dva/ix) within the stream line phase, are 
practically independent of roughness, at least so far as stream line 
flow prevails. There is, however, some evidence that the critical 
value for rough pipe occurs at smaller values of the abscissa {Dvcr/fx) 
than in the case of smooth pipe. On the other hand, for turbulent 
flow the values of / or of ^(Dvcr/fi) vary in marked degree with rough- 
ness. Referring to Fig. 133 which, as noted, shows the results for 
diverse substances with varying diameters and velocities, but all 
with smooth pipe, it is found for rough pipe that the curve beyond the 
critical velocity is of the same general form as for smooth but, with 
increasing roughness, located higher and higher and with increasing 
approach to parallelism with the axis of abscissa. This approach of / 
to a constant value implies, as previously noted, a corresponding 
approach to a law of variation of Gorf with the square of the speed. 

For ordinary iron or steel pipe, plain or galvanized, and with varying 
degrees of roughness as met with in actual practice, the values of / 
will range between those indicated in Table XXXII for smooth or 
new pipe, up to values 50 to 100 per cent greater for very rough and 
pitted pipe surfaces. The influence due to roughness must, therefore, 
be allowed for according to judgment and in accordance with the 
observed or assumed condition of the surface.* 

For values of the abscissa close about the critical state, 2500 to 2000 
or lower for rough pipe, the values of / will be very uncertain, due 
apparently to unsteady conditions of flow involving the irregular 
formation and disappearance of eddies and turbulence, 

* Compare also discussion of values of / in Sec. 7. 



APPENDIX I 



261 



The entire field close about the critical value needs further examina- 
tion, especially as to the dependence of the critical value upon rough- 
ness of pipe and, possibly, density of liquid. 

It should be noted that in all usual cases of handling water, at least 
on an engineering scale, the conditions are such as to determine 
turbulent flow. This is readily verified by substituting usual numerical 
values in the abscissa (Dvor/fJi). On the other hand, with many liquids 
handled in industry (oils, syrups, etc.) the conditions are frequently 
such as to determine stream line flow. 



TABLE XXXII 



Lbscissa 


/ 


Abscissa 


/ 


200 


•3200 


14,000 


•0292 


400 


•1600 


16,000 


•0280 


600 


•1067 


18,000 


•0271 


800 


•0800 


20,000 


•0264 


1,000 


•0640 


25,000 


•0249 


1,200 


•0533 


30,000 


•0238 


1,400 


•0457 


35,000 


•0228 


1,600 


•0400 


40,000 


•0219 


1,800 


•0355 


45,000 


•0213 


2,000 


•0320 


50,000 


•0208 


2,500 


•0442 


60,000 


•0200 


3,000 


•0426 


70,000 


•0195 


3,500 


•0412 


80,000 


•0190 


4,000 


•0400 


90,000 


•0185 


4,500 


•0390 


100,000 


•0180 


5,000 


•0382 


150,000 


•0168 


6,000 


•0364 


200,000 


•0158 


7,000 


•0350 


250,000 


•0150 


8,000 


•0340 


300,000 


•0144 


9,000 


•0330 


350,000 


•0140 


10,000 


•0320 


400,000 


•0137 


12,000 


•0304 


450,000 


•0134 



Values of Coefficient / on Abscissa (Dvcr/fx) for smooth 
brass and steel pipe. 

It must not be assumed that the values of / in Table XXXII for 
turbulent flow are the least which may be obtained. It is simply a 
question of smoothness of surface. These values, as noted, refer to 
what may be termed " commercially smooth " pipe. It is possible 
that specially prepared or treated surfaces might show somewhat 
lower values ; and, in fact, there is some evidence tending to indicate 
somewhat lower values in the case of water in new cast-iron pipe, 
asphaltum dipped. The tabular values are to be understood as applying 
to all cases of what may be termed " commercially smooth " surfaces, 
and thus form a general datum from which the effect of roughness 
may be estimated. 

For the general problem of pipe line flow with any fluid whatever 
for which the density and viscosity are known or are determinable, 
it is therefore only necessary to find, for the conditions of operation, 



262 



HYDKAULICS OF PIPE LINES 



the value of the argument Dva/fjL, and thence, guided by judgment 
according to the factor of roughness, to select a suitable value of the 
coefficient /, and thence as in Sec. 5. 

In Table XXXIII are given values of /x the absolute viscosity for 
water at varying temperatures between and 100 C. or 32 to 212 F., 
likewise values of the density for the same temperature ranges. It 
should be especially noted that, with all other factors the same, the 
value of the abscissa {Dva/fi) will vary inversely with the ratio fi/o. 
For handling water under widely varying temperature conditions, 
therefore, the influence of the latter on the value of (Dva/fx) should 
be allowed for in selecting the most appropriate value of /, 



TABLE XXXIII 
Absolute Viscosity and Density of Water 



Temperature 


Viscosity, 


Density 


C. 


F. 


ft., lb., sec. 


ft., lb. 





32 


•001204 


. 62 


42 


5 


41 


•001021 


62 


42 


10 


50 


•000879 


62 


41 


15 


59 


•000766 


62 


38 


20 


68 


•000673 


62 


33 


25 


77 


•000601 


62 


26 


30 


86 


•000538 


62 


17 


35 


95 


•000486 


62 


08 


40 


104 


•000441 


61 


97 


45 


113 


•000402 


61 


85 


50 


122 


•000369 


61 


70 


55 


131 


•000340 


61 


54 


60 


140 


•000315 


61 


37 


65 


149 


•000293 


61 


20 


70 


158 


•000273 


61 


02 


75 


167 


•000255 


60 


83 


80 


176 


•000240 


60 


64 


85 


185 


•000225 


60 


44 


90 


194 


•000213 


60 


22 


95 


203 


•000201 


60 


00 


00 


212 


•000191 


59 


76 



APPENDIX II 



EXPRESSION FOR F CHAPTER III IN TERMS OF v AND e 

Instead of expressing F in a form directly dependent on the quantities 
m and /, it will sometimes be more convenient to have its value in a 
form dependent rather on some steady motion velocity v together with 
the ratio of the actual opening m to the opening m Q for such steady 
motion. 

Put the ratio m/m =e. Then we may assume values of e, such as 
1 0, -9, -8, -7, -6, etc., without knowing or assuming m or m individually. 

From p. 1 14 we have 

(am) 2 

F -TW* nd < ! > 

M "Ws +L ^ (2) 

Using the above value of m in terms of w and e we have 

Af-m'T— i— i + i-l (3) 



r-*-+-*-i 



But from Chapter I equation (45) we readily derive for the present 
oase 

1 _ H L 

2gfm 9 * ~ V C % r (4) 

Substituting this in i[3) we find 



"-"'[.-^-GS-0^] (5) 



and putting this in (1) we have finally, 

(ae) 2 



F =- 
2 



[S-p-^-w] * 6) 



We may then assume that there is some m which gives v . Then, 
whatever it is, we assume that m has a value measured by some fraction 
of this as -8, -6, -4, etc. These fractional ratios then represent values 
of e in the equations above and from (6) with v and e the value of F is 
readily found, 

263 



APPENDIX III 



Proposition : In any hydraulic system or element containing water 
in motion, and where the dimensions are such that we may neglect 
the weight of the water as such, the force reaction of the water on 
the system will be given by the vector sum of the following systems 
of forces : 

(a) The total pressures over the ideal sections bounding the system 
or element, reckoned from without inward and combined as vectors. 

(6) The sum of the momenta per second at inflow and outflow, the 
former taken direct and the latter reversed and all combined as vectors. 

This proposition may be established as follows : 

Consider the hydraulic system of Fig. 91 comprising a pipe AD 
with water flowing through, entering at the section AB and discharging 
at CD. Let p l9 v lt A u p 2 , v 2 , A 2 be respectively the pressure, velocity 
and area at these sections. 

There enters the system in unit of time the volume A-p x with velocity 
v lt and hence the momentum wAjV^/g directed along the line of flow 
at AB. There leaves the system in the same unit of time the momentum 
wA 2 v^jg directed along the line of flow at CD. There is produced, 
therefore, per unit of time, a certain change in momentum. That is, 
under steady flow, there is produced in connection with this system 
a steady rate of change of momentum. 

But a change of momentum is evidence of the operation of a force, 
and the rate of change is the measure of such force. Hence there must 
be in operation on the water, forces or systems of forces of which the 
resultant will be measured by the rate of change of momentum pro- 
duced. 

We now ask what forces or systems of forces can act on the water 
contained within the enclosure ABGD. First, considering that the 
element is substantially in one plane or that its dimensions are such 
that we may neglect the influence due to the weight of the water 
itself, we may classify the remaining forces as follows : 

1. The end forces p±A x and p 2 A 2 acting from without inward and 
normal to the ideal sections A x and A 2 . 

2. The direct force reaction between the inner surfaces of the in- 
closure and the contained water and estimated from the inclosure to 
the water. 

Adding now the rate of change of momentum as the third force 
system involved, we have 

3. The rate of change of momentum produced between AB and CD. 
Then as we have seen, system (3) will be a measure of the resultant 

of (1) and (2). We may express this by the vector equation : 

where S lt S z and S t denote the three systems of forces. 

264 



APPENDIX III 265 

Then by transposition we have : 

Sx— S t = — S 2 . 

Now with the system S 3 , let the entering momentum be denoted 
by M a and the issuing momentum by M b . Then in a vector sense : 

S a =M b -M a 

and-# 8 = -[M b -M a )=M a +[-M b ). 

This shows —S 3 as measured by the vector sum of the momenta 
per second at inflow and outflow, the former \M a ) taken direct and the 
latter (ikf&) reversed. 

Furthermore — S 2 means S 2 reversed : that is, the force reaction 
from the water to the inclosure. 

But this is exactly what is wanted, and the above analysis therefore 
shows that this is measured by the vector sum of S t and ( — ) S 2 , and 
these are made up as in the statement of the proposition. 

The proof is readily extended to include the case with any number 
of points of inflow and any number of outflow. 

If the dimensions are such that the weight of the water cannot be 
neglected, we must then add this as a fourth system acting vertically 
downward through the centre of volume of the element. Denoting 
this by Si we shall then have as the vector equation for this case : 

$i +£4 — £«= —S t . 
This equation expresses ( — ) S 2 (the force reaction desired) as the 
vector sum of three vector systems specified and defined as above. 



APPENDIX IV 



ECONOMIC DESIGN 

In the case of all carriers of energy in its various forms, whether pipe 
lines for water, steam or air, or metal lines for electricity, the same 
fundamental problem of economic design presents itself. 

Broadly speaking the annual cost chargeable against such a line 
arises under two heads. 

1. Fixed charges proportional generally to investment or to first 
cost. 

2. Operating costs, resulting from the annual operating program. 
Let X and Y denote respectively these two classes of cost and u 

the total. Then u=X + Y . . (1) 

Now in general it will result that a change in the size of the carrier 
(pipe or wire) will affect X and Y in opposite directions. Thus an 
increase in the size will increase the cost and hence the fixed charges 
while it will decrease, in general, the secondary losses (friction or 
electrical resistance) and thus decrease the operating costs for the 
same energy carried. 

Similarly a decrease in the size of the carrier will produce changes 
in the opposite direction. It thus results that there may well be some 
value of the size of the carrier for which the total cost X + Y will be a 
minimum. To investigate such a possibility we proceed in the usual 
manner. Thus let x denote in general diameter or size. Then we 

haVe ^L = ^+^I (2) 

dx dx dx' ' 

also — = — +— (3) 

dx*~ dx 2 dx 2 ' 

For a maximum or minimum dy/dx=0 and hence 

dX = _dI | 4) 

dx dx 

Likewise for a minimum, d*ujdx* is positive in sign. 

These general conditions are indicated geometrically in Fig. 134, 
where XX and YY denote the general character of the curves for X 
and Y plotted on x or size. 

Then the condition of equation (4) may be put into words as follows : 

A maximum or minimum value of u will be found for the value of 
x, for which the slopes X and Y are numerically the same, but opposite 
in direction. This will be at some point A in the curves of the diagram. 

Again, to determine under what conditions this point will correspond 
to a minimum we have only to inquire as to the sign of d z u IdxK 

266 



APPENDIX IV 



267 



The sign of d x Xldx i or d*Y '/dx* is determined by the direction of 
rotation of the tangent for increasing x. This will be -f for counter 
clockwise rotation and — in the inverse case. In the curves of Fig. 
134, convex to the axis of x, it is seen that in both cases as x increases 
the tangent to the curve will rotate counter clockwise. Hence both 
d t X/dx z and d z Y/dx 2 are + and the sum will be + , and hence d*u/dx 2 
will be plus and the point determined as the sum of AP+AQ will 
be a minimum. If either of these lines is straight, the second derivative 
is zero, but that of the other will be + and hence the sum will be plus 




.r 
Fig. 134. — Economic Value, Graphical Determination. 



and the point determined by (4) will give a minimum value. The 
reader will readily extend this analysis to forms which are concave 
to the axis of x and which will correspondingly determine a maximum 
for the sum of X and Y. 

In all usual cases, however, in which pipes or wires are used as 
carriers of energy, one or both of the curves will be convex to the axis 
of x and hence the condition of (4) will determine a minimum. 

As to the actual determination of the point A, various methods are 
open. If the two curves are plotted individually, an easy trial and error 
test will serve to find the points P and Q on the same ordinate where 
the slope is the same but in opposite directions. 

Again, the sum of X and Y may be plotted, as indicated in the curve 
UU, and the minimum point determined by inspection. 

Again, from (4) we have 

*[— 1. 

dX 

Hence if Y and X are taken as the axes, or otherwise if we plot 
X on F, the values of X and Y which will produce the minimum value 
of u will be where the slope of the resulting curve is 135°, or where it is 
45° with the horizontal. See construction in dotted lines. 



INDEX 



Air relief valves, 234 

Allievi's formula for excess pressure 

due to water-ram, 151 
Angles and bends, loss of head due to, 

23 

B 

Bellasis, loss of head due to con- 
traction, 20 

Bends and angles, loss of head due to, 
23 

Breaking plates, 240 

Brightmore, loss in elbows and 
bends, 26 

Bulkheads, 243 

Buried and unburied pipe, relative 
advantages of, 231 

C 

Calking of riveted seams, 202 
Capacity, general formula for, 38 

round pipe running partly full, 
39 
Cast-iron pipe, 192 
Chezy coefficient, practical values of, 
8 
coefficient values from ship re- 
sistance experiments, 15 
formula, 4 
Church's solutions surge chamber 

problem, 69 
Commercial pipe, 191 
Concrete (reinforced) pipe, 211 
Construction, 190 
Contraction, abrupt, loss of head due 

to, 19 
Cross section of pipe, distribution of 
velocity over, 32 
mean velocity over, 33 

D 

Darcy's formula, 7 

Davis, loss in elbows and bends, 27 

Density of oil, 247 

Design, 211 

of oil pipe lines, 253 



Diameter of line, economic size, 212 
Differential surge chamber, 83 

E 

Economic design, general principles, 
265 

diameter of line, 212 
Elbows, loss of head due to, 23 
Electric welding in lieu of calking, 203 
Energy of flowing stream, 1 
Engler viscosimeter, 250 
Entrance, loss of head at, 18 
Erection of steel pipe lines, 227 
Expansion, abrupt, loss of head due 
to, 19 

and contraction in pipe lines due 
to temperature changes, 225 

joints, 242 
Exponential formula, 6 



F 



17 



Fifth powers of numbers 
Fittings, pipe line, 245 
Flowing stream, energy of, 1 
Free surface flow, 44 

G 

Gradient, hydraulic, 35 
Gravity of oil, 247-250 



H 

Head, friction, 2-30 

friction in pipe made up of 

sections of different diameters, 

16 
loss of, 2-30 

loss of, general resume^ 29 
total, 1 
Hydraulic conditions in riveted pipe 

lines, 201 
gradient, 35 



Johnson's formula surge chamber 

problem, 73 
Joint, cast-iron pipe, 192 



269 



270 



HYDRAULICS OF PIPE LINES 



Jointa and connections (steel pipe), 

203 
Joukovsky's formula for excess 

pressure due to water-ram, 154 

K 

Kuiohlinq and Smith, loss through 

valves, 22 
Kutter's formula, 5 



Larner's formula surge chamber 
problem, 74 

'Loss of head at entrance, 18 

due to angles and bends, 23 

due to friction, 2-30 

due to obstruction, 20 

due to sudden contraction, 19 

due to sudden expansion, 19 

general resume, 29 

M 

Manholes and covers, 242 
Materials, 190 

Merriman's formula, loss due to con- 
traction, 20 

O 

Obstruction, loss of head due to, 20 
Oil pipe lines, 246 



Paint for pipe lines, 232 

Piers and anchors, 228 

Pipe -line connecting two reservoirs, 

47 
flow, general theory of, 257 
Piping systems, 51 
Power delivered at discharge end of 

line, 50 
Pressure relief valves, 240 
Protective coatings, 232 

R 

Rankine's formula, loss due to 

obstruction, 21 
Redwood viscosimeter, 250 
Relief valves, 234 
Riveted joints, calking of, 2Q2 

joints in sheet steel pipe, circum- 
ferential, 199 
joints in sheet steel pipe, longi- 
tudinal, 195 
pipe lines, hydraulic conditions 
in, 201 



S 

Saybolt viscosimeter, 250 
Schoder, loss in elbows and bends, 26 
Shock, see water-ram, 84 
Smith, Hamilton-coefficients, 12 
Steady flow, general problem of, 41 
Steel (sheet) pipe, 194 
Stresses in combinations of bends or 
elbows with expansion joints, 
174 
in connections and fittings, 184 
in expansion joints, 173 
in flat plates, 187 
in joint fastenings, flanges, bolts, 

etc., 188 
in long pipe with open ends 

carried in slip joints, 176 
in supporting ribs, 187 
Stresses in pipe lines, 159 
combined, 189 
due to angles, bends and fittings, 

161 
due to bending moment in spans, 

188 
due to expansion and con- 
traction, 226 
including load due to weight of 
pipe or element with contained 
water, 172 
influence of anchors, piers, etc., 

180 
longitudinal stress, 160 
ring tension, 1 60 
Surge chamber equations, treatment 
of, 68 
general statement of problem 
and derivation of equations, 59 
Surge Chamber Problem, simplified 
by disregard of governor 
action and assuming friction 
head to vary with first power 
of velocity, 75 
treatment by model experiment 
through application of law of 
kinematic similitude, 79 
treatment by numerical integra- 
tion, 77 
treatment through assumption 
of predetermined program of 
acceleration, 79 



Test flanges, 243 

Thickness, cast-iron pipe, deter- 
mination of, 193 
sheet steel pipe, determination 
of, 219 



INDEX 



271 



Valves, loss of head through, 22 

pipe -line, 244 
Vensano's formula for excess pressure 

due to water-ram, 155 
Viscosimeters, 250 
Viscosity of oil, 248 

W 
Warren's formula for excess pres- 
sure due to water-ram, 153 
Water-ram, 84 

Allievi's formula, 151 
approximate formulae, 150 
discussion of formulae, with 

numerical cases, 128 
excess pressure developed, 97 
formulae based on mass and 

acceleration, 155 
Joukovsky's formula, 154 
law of pressure change with time, 

closure, 107 
law of pressure change with time, 

opening, 110 
physical conditions necessary 
to allow for elasticity of pipe, 

friction and velocity head, 89 
velocity of propagation of 

acoustic wave, 93 



Water-ram — 

Vensano's formula, 155 

Warren's formula, 153 

when lower end of pipe is held 

rigid, 105 
with gradual complete closure, 

110 
with gradual partial closure, 121 
with gradual opening, 125 
with instantaneous complete 

closure, 84 
with partial reflection at valve, 

closure, '122 
with partial reflection at valve, 

opening, 127 
with rapid opening from com- 
plete closure, 106 
with rapid opening from partial 

initial opening, 107 
with rapid complete closure, 98 
with rapid partial closure, 102 
Weisbach's results, loss due to con- 
traction, 20, 24 
Welded sheet steel pipe, 198 
Williams and Hazen's formula, 

friction loss, 6 
Williams Hubbell and Fenkell, loss in 

elbows and bends, 25 
Wood stave pipe, 205 



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